Combinatorics Study Group

The Tutte polynomial of a matroid has a formula arising from algebraic
geometry; surprisingly given the context, it also works for
non-realisable matroids. The formula reveals its close kinship with
some other invariants: the Ehrhart polynomial and an invariant of
Speyer notionally counting the pieces in a decomposition into
series-parallel matroids. In fact the algebraic geometry needed can
be worked with in a wholly combinatorial fashion, using splines and
lattice point enumeration in cones. I'll explain this, beginning from
a review of the matroid notions.

This work is joint with David Speyer.

Given a tree T with n vertices, how large does p need to be for it to be likely that a copy of T appears in the binomial random graph G(n,p)? I will discuss this question, including recent work confirming a conjecture which gives a good answer to this question for trees with bounded maximum degree.

A point-line framework is a collection of points and lines in the Euclidean plane which are linked by constraints which fix the angles between some pairs of lines, and the distances between some pairs of points and between some pairs of points and lines. It is rigid if the only continuous motion of the points and lines which preserve the constraints are translations or rotations of the whole plane. The rigidity of a framework depends only on its underlying `point-line graph' when the framework is generic i.e there are no algebraic dependencies between the coordinates of its points and lines. We characterize when a generic point-line framework is rigid. This is joint work with John Owen.

The commuting probability of a finite group is defined to be the probability that two randomly chosen group elements commute. Not all rationals between 0 and 1 occur as commuting probabilities. In fact Keith Joseph conjectured in 1977 that all limit points of the set of commuting probabilities are rational, and moreover that these limit points can only be approached from above. In this talk we'll discuss a structure theorem for commuting probabilities which roughly asserts that commuting probabilities are nearly Egyptian fractions of bounded complexity. Joseph's conjectures are corollaries.

Graph Isomorphism is one of the classical combinatorial problems whose complexity status is wide open. In my talk, I will focus on graphs of bounded treewidth, and present known approaches to tackle this problem in these graph classes. In particular, I will present the recent fixed-parameter tractable algorithm for Graph Isomorphism parameterized by the treewidth that relies on an isomorphic-invariant modification of the well-known Robertson-Seymour approximation algorithm for treewidth. The talk is mostly based on http://arxiv.org/abs/1404.0818(link is external), presented at FOCS 2014.

Let G be a multigraph without loops. A Tutte trail of G is defined to be a trail H in G such that each component of G\V(H) has at most three edges connecting it to H. In this talk, I will show that a 2-edge-connected plane graph has a Tutte trail. I will also discuss the relationship between Tutte trails and the conjecture of Carsten Thomassen that every 4-connected line graph is Hamiltonian.

The Ehrhart polynomial of a lattice polytope P is the enumerative function counting the number of lattice points contained in the dilated polytope nP for any positive integer n. In this talk, after introducing the fundamental facts on Ehrhart polynomials of lattice polytopes, I'll present some characterizing results on them.

In this talk we will discuss the rigidity and flexibility of bar-joint frameworks in (d+1)-dimensional space whose vertices are constrained to lie on concentric d-spheres with independently variable radii. In particular, we present combinatorial characterisations of generic rigid frameworks for d=1 with an arbitrary number of independently variable radii, and for d=2 with at most two variable radii. This includes a characterisation of the rigidity or flexibility of uniformly expanding spherical frameworks in 3-space.

Due to the equivalence of the generic rigidity between Euclidean space and spherical space, these results interpolate between rigidity in 1D and 2D and to some extent between rigidity in 2D and 3D.

Our initial motivation for this work was to understand the spherical mechanisms of mathematical toys inspired by popular structures like the Hoberman sphere (such as Juno's spinners, for example). Since many of these structures exhibit non-trivial symmetries, we also briefly discuss the impact of symmetry on the flexibility of frameworks on variable spheres.

This is joint work with Anthony Nixon, Shin-ichi Tanigawa and Walter Whiteley.

In 1997, Bollobás and Leader gave tight lower bounds for the number of edge-disjoint paths between disjoint subsets A and B of the hypercube, Q_n, in terms of |A| and |B|. They conjectured that this bound holds even when A is a down-set, B is an up-set and the paths are required to be directed (that is, the vertices in the path form a chain). I will sketch a novel compression-type argument that proves a stronger version of this conjecture.

Given two probability distributions P_R and P_B on the positive reals with finite means, colour the real line alternately with red and blue intervals so that the lengths of the red intervals have distribution P_R, the lengths of the blue intervals have distribution P_B, and distinct intervals have independent lengths. Now iteratively update this colouring of the line by coalescing intervals: change the colour of any interval that is surrounded by longer intervals so that these three consecutive intervals subsequently form a single monochromatic interval. Say that a colour (either red or blue) `wins' if every point of the line is eventually of that colour. This talk will attempt to answer the following question: under what natural conditions on the distributions is one of the colours almost surely guaranteed to win?

Recent works of Kodama et. al. (Chakravarty and Kodama 2009, Kodama and Williams 2013) have demonstrated deep connections between soliton solutions of the Kadomtsev-Petviashvili (KP) Equation and well-known combinatorial structures, including generating polynomials, Grassmann necklaces, Young diagrams, and matroids.

Joint work with A. Redlich (Bowdoin Univ.) provides a simple extension to the class of KP solitons which may be considered in this context. More recently, progress has also been made with A. Hamm (Winthrop Univ.) in understanding some of these structures for the Jimbo-Miwa equation. In light of this "modern combinatorics toolkit", we present several open questions on what may make a Wronskian-integrable PDE admit (or exclude) certain combinatorial structures.

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If a triangulation of the sphere has only two vertices of odd degree, then these cannot be adjacent. We prove this curious fact by studying the colouring monodromy, an obstruction to the existence of a vertex colouring. The monodromy also leads to a construction of equivelar surfaces of high genus.

Similar in the spirit is the following theorem. There is no triangulation of the torus with all vertex degrees 6, except for one of degree 5 and one of degree 7. This is proved by studying the rotation monodromy of the geometric structure obtained by making each triangle of the triangulation to an equilateral euclidean triangle.
The second result is a joint work with Kusner, Rote, Springborn, and Sullivan.

We study the properties of finite graphs in which the ball of radius r around each vertex induces a graph isomorphic to some fixed graph F. (Such a graph is said to be r-locally-F.) This is a natural extension of the study of regular graphs, and of the study of graphs of constant link. We focus on the case where F is L^d, the d-dimensional integer lattice. We obtain a characterisation of all the finite graphs in which the ball of radius 3 around each vertex is
isomorphic to the ball of radius 3 in L^d, for each integer d. These graphs have a very rigidly proscribed global structure, much more so than that of (2d)-regular graphs. (They can be viewed as quotient lattices in certain 'flat orbifolds'.) Our results are best possible in the sense that '3' cannot be replaced with '2'. Our proofs use a mixture of techniques and results from combinatorics, algebraic topology and group theory. Joint work with Itai Benjamini (Weizmann Institute of Science).

We prove that for every poset P, there is a constant C_P such that the size of any family of subsets of [n] that does not contain P as an induced subposet is at most C_P (n \choose n/2), proving a conjecture of Katona, and Lu and Milans. We obtain this bound by establishing a connection to the theory of forbidden submatrices and then applying a higher dimensional variant of the Marcus-Tardos theorem, proved by Klazar and Marcus.

A two-dimensional direction-length framework (G, p) consists of a multigraph G = (V ; D, L) whose edge set is partitioned into “direction” edges D (which fix the gradient of the line through both end-vertices), and “length” edges L (which specify the distance between their end-vertices); together with a realisation p of this graph in the plane.

Given a direction-length framework (G, p), we want to know whether there are other realisations of G which satisfy the same direction and length constraints. Any such framework (G, q) is said to be equivalent to (G, p)
We can easily obtain frameworks which are equivalent to (G,p) by either translating (G, p) in the plane or rotating it by 180 degrees, but are these the only equivalent frameworks? If so we say that (G,p) is globally rigid.

Characterisations of global rigidity exist for two classes of generic frameworks, but we do not yet have a full characterisation. In this talk I will explain the methods used to obtain these two characterisations, and describe to what extent these methods might be helpful in tackling the remaining cases.

We start with a brief introduction to chip firing games. We then discuss an application of chip firing games to a problem in combinatorial commutative algebra, namely explicit deformations of lattice ideals. This talk is based on joint work with Spencer Backman.

The \emph{$r$-neighbour bootstrap process} on a graph $G$ starts with an initial set of ``infected'' vertices and, at each step of the process, a healthy vertex becomes infected if it has at least $r$ infected neighbours (once a vertex becomes infected, it remains infected forever). If every vertex of $G$ becomes infected during the process, then we say that the initial set \emph{percolates}.

In this talk I will discuss the proof of a conjecture of Balogh and Bollob\'{a}s: for fixed $r$ and $d\to\infty$, the minimum cardinality of a percolating set in the $d$-dimensional hypercube is $\frac{1+o(1)}{r}\binom{d}{r-1}$. One of the key ideas behind the proof exploits a connection between bootstrap percolation and weak saturation. This is joint work with Jonathan Noel.

Let $T$ be a tile in say $\mathbb{Z}^2$, meaning a finite subset of
$\mathbb{Z}^2$. It may or may not be the case that $T$ tiles
$\mathbb{Z}^2$, in the sense that $\mathbb{Z}^2$ can be partitioned
into copies of $T$. But is there always some higher dimension $d$ such
that $T$ tiles $\mathbb{Z}^d$? We prove that this is the case: for any
tile in $\mathbb{Z}^2$ (or in $\mathbb{Z}^n$, any $n$) there is a $d$
such that $\mathbb{Z}^d$ can be tiled with copies of it. This proves a
conjecture of Chalcraft. Joint work with Imre Leader and Ta Sheng Tan.

An F-decomposition of a graph G is a partition of E(G) into copies of F. Determining whether a graph has an F-decomposition is NP-complete, but it is much easier to find 'fractional' decompositions. I'll explain the connection between these ideas and how it can be exploited to attack Nash-Williams' conjecture that every large graph on n vertices with minimum degree at least 3n/4, e(G) divisible by 3 and every degree even, has a triangle-decomposition. I'll also mention an application to completion of partial mutually orthogonal latin squares.

I will discuss a construction of finite 'geometric' random
graphs motivated by the study of random walks on infinite groups. This
construction has connections to other topics, including the Poisson
boundary and Sznitman's random interlacements (which I will try to
introduce in a gentle way).

http://arxiv.org/abs/1506.02697

Given a graph $G$, we define $\lambda_1(G)$ to be the largest eigenvalue of the adjacency matrix of $G$. We consider the following isoperimetric question: among subgraphs of the cube $Q_d = \{0, 1\}^d$ of prescribed order, which graphs maximise $\lambda_1$?

We believe that in most cases Hamming balls are maximisers and our result support this: we prove that Hamming balls are asymptotically best when the radius is $o(d)$; furthermore, when the radius is fixed, Hamming balls are maximisers for large $d$. Compressions play an important role in our proofs.

This is joint work with Jonathan Lee and B\'ela Bollob\'as.

Bill Cook's lecture In pursuit of the traveling salesman: mathematics at the limits of computation is liable to be of interest to the CSG audience.

Erdos asked the following question: given a positive integer n, what is the largest integer k such that any set of n points in a plane, with no 4 on a line, contains k points no 3 of which are collinear? Furedi proved that k = o(n). Cardinal, Tooth and Wood extended this result to R^3, finding sets of n points with no 5 on a plane whose subsets with no 4 points on a plane have size o(n), and asked the question for the higher dimensions. For given n, let k be largest integer such that any set of n points in R^d with no more than d + 1 cohyperplanar points, has k points with no d + 1 on a hyperplane. Is k = o(n)? We prove that k = o(n) for any fixed d \geq 3.

Given a family of graphs $\mathcal{H}$, the extremal number $\textrm{ex}(n, \mathcal{H})$ is the largest $m$ for which there exists a graph with $n$ vertices and $m$ edges containing no graph from the family $\mathcal{H}$ as a subgraph. We show that for every rational number $r$ between $1$ and $2$, there is a family of graphs $\mH_r$ such that $\textrm{ex}(n, \mH_r) = \Theta(n^r)$. This solves a longstanding problem in the area of extremal graph theory.

Joint work with Boris Bukh.

In this talk we present variants of two classical extremal problems:
estimating Ramsey numbers for cliques and Tur\'an numbers for complete
bipartite graphs. Our results (obtained jointly with Conlon and Fox)
are proved by basic probabilistic arguments and answer questions of
Bollob\'as, Erd\H{o}s, Foucaud, Hajnal, Krivelevich and Perarnau.

In this talk, I will present a family of graphs that represent
particle interactions in a field theory. They also represent
Grassmannians. Using matroid technology, I answered the physically
interesting question of which diagrams correspond to positive
Grassmannians. I also discuss open questions regarding defining a
differential operator, combinatorially, on this family of graphs.

Rado showed that the random graph with a countable vertex set
and each chosen independently with probability $1/2$ is, almost surely,
unique up to isomorphism. Bonato and Janssen showed that there are some
countable random geometric graphs with the same property, but that this
depended on the underlying geometry (i.e., the norm on the underlying
normed space). They also gave a partial classification of which norms in
two dimensions do give a unique graph.

We give a complete classification of all norms in all (finite)
dimensions. The proof mixes combinatorics, probability and the geometry
of finite dimensional lattices.

This is joint work with Paul Balister, B\'ela Bollob\'as, Karen
Gunderson and Imre Leader.

Motivated by a sampling problem arising in statistics, Diaconis, Graham and Holmes
introduced a simple Markov chain, the switch chain, on the set of all perfect
matchings in a bipartite graph G. Two basic questions about this Markov chain are:
for which class of graphs is the switch chain ergodic, and for which is it rapidly
mixing? (I.e., does the switch chain have a limiting distribution and, if so,
what is its rate of convergence?) I'll present a precise answer to the ergodicity
question and close bounds on the mixing question. In particular, I'll give some
rough indications of a proof that the mixing time of the switch chain is polynomial
in the size of the graph G in the case that G is monotone (a.k.a. a bipartite
permutation graph). The class of monotone graphs includes examples of interest
in the statistical setting.

Let A and B be two partially ordered sets. We say that B contains A
strongly (or: "inducedly") if there exists an injective map f from A to
B such that f(x) < f(y) if and only if x < y for every x, y in A. In
this talk, I will prove that for every (finite) partially ordered set P,
there exists a constant c(P) such that if a subset F of the hypercube
(ordered by inclusion) does not contain P strongly, then the LYM density
of F is at most c(P), where the LYM density of F is defined to be the
sum of the densities of F within each level of the hypercube. This
proves a conjecture of Lu and Milans, and strengthens a result of
Methuku and Pálvölgyi.

Recently there has been a lot of chatter from various parties about
generalising Conway's extraordinary construction of the small Mathieu
groups from projective planes. For some reason several of these people
seem to think I should put my oar in and so here are some thoughts on
these recent constructions along these lines and potential avenues for
generalisations.

The well-known colouring number of a graph is one more than the smallest
k for which there exists an ordering of the vertices in which each
vertex has at most k neighbours that come earlier in the order. The
colouring number is a trivial upper bound for the chromatic number of
the graph. But it also contains structural information of the graph, for
instance regarding the edge density of any subgraph.
When instead of neighbours that are earlier in the order we consider
vertices that are earlier and can be reached by a path of length at most
r, we get the generalised colouring numbers. Depending on where we want
the internal vertices of such paths to appear in the order, we actually
can define different types of those generalised colouring numbers.
Generalised colouring numbers were introduced by Kierstead and Yang in
2003, to study specific types of graph colourings. In the last couple of
years it has been realised that the generalised colouring numbers are
closely related to many structural properties of graphs (such as
tree-depth and tree-width).
In this talk we give an overview of the relations between the
generalised colouring numbers and specific types of colourings of
graphs; we discuss relations between those numbers and structural
properties of graphs; and we discuss some new bounds on these numbers
for specific classes of graphs (such as planar and minor-closed graphs).

Joint work with Patrice Ossona de Mendez, Daniel Quiroz, Roman
Rabinovich, and Sebastian Siebertz.

A theorem of Lafforgue asserts that a matroid whose polytope cannot be
dissected into smaller matroid polytopes can have only finitely many
representations over any field, and in particular no continuously
deformable representation. We explain a proof using the machinery of
valuated matroids, which we present by way of hyperfields following
Matt Baker's recent work.

Given some dimension $d$, what is the maximum number of lines in $\mathbb{R}^d$ such that the angle between any pair of lines is constant? This classical problem has recently enjoyed a renewed interest due to the current attention the quantum information community is giving to its complex analogue. I will report on some new developments of the theory of equiangular lines in Euclidean spaces. Among other things, I will present a new construction using real mutually unbiased bases and improvements to two long standing upper bounds for equiangular lines in dimensions 14 and 16.

An alternating sign matrix (ASM) is a square matrix in which each entry is -1, 0 or 1, and along each row and column the nonzero entries alternate in sign, starting and ending with a 1. It was conjectured by Mills, Robbins and Rumsey in 1982 that the number of ASMs of fixed size is given by a certain simple product formula. A relatively short proof of this conjecture was obtained by Kuperberg in 1996, using connections with the six-vertex model of statistical mechanics. It was also conjectured by Robbins in the mid 1980's that the number of ASMs of fixed odd size which are invariant under diagonal and antidiagonal reflection is given by a simple product formula. This conjecture has only recently been proved, again using connections with the six-vertex model, in my joint work with Ilse Fischer and Matjaz Konvalinka (see arXiv:1512.06030). In the first part of this talk, I'll introduce ASMs, and review Kuperberg's proof of the ASM enumeration formula. In the second part, I'll outline the proof of the enumeration formula for diagonally and antidiagonally symmetric ASMs of odd order.

Informally, a body-bar framework consists of some rigid bodies and some bars each connecting two different rigid bodies. We can use multigraphs to illustrate body-bar frameworks by adding a vertex for each rigid body
and adding k edges between two vertices if there are k bars connecting the rigid bodies corresponding to these vertices.

Tay showed that a generic body-bar framework is rigid if and only if the underlying multigraph contains the union of six edge-disjoint spanning trees, where a framework is generic if the endpoints of the bars are algebraically independent over the rationals. We will summarise Tay's result and then introduce some classes of non-generic
body-bar frameworks that have the same characterisation as the generic ones.

Many of the fastest known algorithms for factoring large integers rely on finding subsequences of randomly generated sequences of integers whose product is a perfect square. Motivated by this, in 1994 Pomerance posed the problem of determining the threshold of the event that a random sequence of $N$ integers, each chosen uniformly from the set $\{1,\dots,x\}$, contains a subsequence, the product of whose elements is a perfect square. In 1996, Pomerance gave good bounds on this threshold and also conjectured that it is {\em sharp}.

In a paper published in Annals of Mathematics in 2012, Croot, Granville, Pemantle and Tetali significantly improved these bounds, and stated a conjecture as to the location of this sharp threshold. In recent work, we have confirmed this conjecture. In my talk, I shall give a brief overview of some of the ideas used in the proof, which relies on techniques from number theory, combinatorics and stochastic processes. Joint work with B\'ela Bollob\'as and Robert Morris.

Classical linear inequality systems can be examined well by linear programming. The tropical counterpart is more difficult to handle.

We examine the problem from a combinatorial point of view related to tropical oriented matroids. Furthermore, we show connections to scheduling and games.

We'll present buildings as universal covers of certain CAT(0) complexes. Fundamental groups of these complexes will be used for expander constructions and for generating an infinite families of tessellated compact hyperbolic surfaces with large isometry groups.

We show that the mixing time of Glauber (single edge update) dynamics for the random cluster model at q = 2 is bounded by a polynomial in the size of the underlying graph. As a consequence the Swendsen-Wang algorithm for the ferromagnetic Ising model at any temperature has the same polynomial mixing time bound.

Joint work with Mark Jerrum.

A family of sets is said to be intersecting if any two sets in the family have nonempty intersection. Families of sets subject to various intersection conditions have been studied over the last fifty years and a common feature of many of the results in the area is that the extremal families are often quite asymmetric. Motivated by this, Peter Frankl conjectured in 1981 that symmetric intersecting families must generally be very small; more precisely, Frankl conjectured that if a family of subsets of {1, 2, . . . , n} with the property that any three sets in the family intersect has a transitive automorphism group, then the family must have size o(2^n). In this talk, I shall prove this conjecture.

Joint work with David Ellis.

A Hamilton cycle in a directed graph G is a cycle that passes through every vertex of G. A Hamiltonian decomposition of G is a partition of its edge set into disjoint Hamilton cycles. In the late 60s Kelly
conjectured that every regular tournament has a Hamilton decomposition. This conjecture was
recently settled by Kühn and Osthus, who proved more generally that every r-regular n-vertex oriented graph G (without antiparallel edges) with r=cn for some fixed c>3/8 has a Hamiltonian decomposition, provided
n=n(c) is sufficiently large.

In this talk I will address the natural question of estimating the number of such decompositions of G. Our main result shows that this number is n^{(1-o(1))cn^2}. As a by product, we also obtain a new and much simpler proof for the approximate version of Kelly's conjecture.

Joint work with Asaf Ferber and Benny Sudakov.

A Hamilton cycle in a directed graph G is a cycle that passes through every vertex of G. A Hamiltonian decomposition of G is a partition of its edge set into disjoint Hamilton cycles. In the late 60s Kelly
conjectured that every regular tournament has a Hamilton decomposition. This conjecture was
recently settled by Kühn and Osthus, who proved more generally that every r-regular n-vertex oriented graph G (without antiparallel edges) with r=cn for some fixed c>3/8 has a Hamiltonian decomposition, provided
n=n(c) is sufficiently large.

In this talk I will address the natural question of estimating the number of such decompositions of G. Our main result shows that this number is n^{(1-o(1))cn^2}. As a by product, we also obtain a new and much simpler proof for the approximate version of Kelly's conjecture.

Joint work with Asaf Ferber and Benny Sudakov.

We will consider a deceptively simple question formulated by Po-Shen Loh concerning sequences of integer triples. We shall discuss some recent developments following joint work with Tim Gowers, and mention a collection of generalisations and open problems.

This is a work-in-progress talk. I will introduce a class of random geometric graphs where each node is associated to a real (scalar or vectorial) value, and the linking probability depends on a
generalised visibility criterion. We will discuss a few basic properties of such graphs in the specific case of uniform point distribution with uniformly sampled node values, and we will provide a review of several interesting open problems and potential applications.

A d-dimensional bar-and-joint framework is said to be globally rigid if every d-dimensional framework with the same underlying graph and with the same edge lengths is congruent to it. In dimensions 1 and 2 global rigidity for generic frameworks can be characterised purely by properties of the underlying graph. In this talk I will describe these characterisations and discuss extensions to frameworks in 3-dimensions whose vertices are restricted to move on a fixed surface. In particular when the surface is a generic family of concentric cylinders I will give a complete description of the graphs whose frameworks are generically globally rigid.

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We consider a degree proportional percolation model in which an infection spreads through a graph by infecting any vertex which has a fixed proportion of its neighbours already infected. This talk will present the background and solution to the problem of identifying the size of a minimal initial target set which will spread the infection to the entire graph.

Joint work with Richard Mycroft and Frederik Garbe.

If S is a subset of {0,1}^n, i.e. S is a set of vertices of the discrete n-
dimensional cube, the 'edge boundary' of S is the set of edges of the cube
which join a vertex in S to a vertex outside S. The edge-isoperimetric inequality
of Harper, Bernstein, Lindsay and Hart specifies, for each integer k, the smallest
possible size g_n(k) of the edge-boundary of a set of k vertices in {0,1}^n; the
extremal sets are initial segments of the lexicographic ordering on {0,1}^n (up
to automorphisms of the discrete cube), e.g. subcubes, when k is a power of 2.
There are several 'stability results' describing the structure of subsets of the
discrete cube with 'small' edge-boundary; some, such as Friedgut's Junta
theorem, have found many applications. We discuss a new stability result,
showing that if a subset of {0,1}^n of size k has edge-boundary of
size at g_n(k) + m, then it can be changed into an extremal set by at most O(m) additions and deletions.
We conjecture that O(m) could be replaced by 2m. Joint work with
Nathan Keller (BIU) and Noam Lifshitz (BIU).

The clique complex of a graph $G$, denoted $X(G)$, is the (abstract) simplicial complex where the vertex set of each clique of size $k$ corresponds to a $(k-1)$--face. For $G=G_{n,p}$, M. Kahle conjectured a precise threshold value $p_0$ so that if $p>p_0$, then a.s. the $k^{\textrm{th}}$ homology group of $X(G)$ over a field $\Gamma$ vanishes. In this talk I will discuss our proof that Kahle's conjecture holds for $k=1$ and $\Gamma=\mathbb{Z}_2$. By regarding $E(G)$ as a $\mathbb{Z}_2$-vector space in the obvious way, the problem reduces to showing that a.s. the triangle space equals the cycle space for $p$ bigger than the conjectured $p_0$ . If time permits, generalizations to higher homology groups will be discussed. Joint work with B. DeMarco and J. Kahn.

Suppose H is a fixed graph. H-colourings of a graph G (a.k.a. homomorphisms from G
to H) generalise familiar proper vertex colourings of G. More than 15 years ago, Dyer
and Greenhill considered the computational complexity of counting H-colourings,
and demonstrated a dichotomy, in terms of the graph H, between polynomial time
and #P-complete.

That result was for exact counting, and, even now, there is only a partial complexity
classification for approximate counting. However, the classification problem becomes
tractable if we look instead at _list_ H-colourings. In this talk, I’ll present a classification
(in fact a trichotomy) for the problem of approximately counting list H-colourings of
a graph. It turns out that some interesting hereditary graph classes come into play
in describing and proving the trichotomy result.

This is joint work with Andreas Galanis and Leslie Ann Goldberg (Oxford).

We define a random geometric graph by choosing n points in a square and joining pairs of points if they are close to each other. It is natural to ask when standard graph properties (such as connectedness, Hamiltonicity, et cetera) typically occur in this model. We consider the property of containing the square of a Hamilton cycle. Our main result is that for a typical point set the graph contains the square of a Hamilton cycle exactly when a simple local condition is satisfied at every vertex. Perhaps surprisingly, this property exhibits quite different behaviour in the binomial random graph. Furthermore, unlike in the case of connectedness and Hamiltonicity, the local condition is not simply a minimum degree condition.

The fundamental problem of distance geometry asks to find a realization of a finite, but partially specified, metric space in a Euclidean space of given dimension. Unless some structure is known about the structure of the instance, it is notoriously difficult to solve these problems computationally, and most methods will either not scale up to useful sizes, or will be unlikely to identify good solutions. We propose a new heuristic algorithm based on a semidefinite programming formulation, a diagonally-dominant inner approximation of Ahmadi and Hall's, a randomized-type rank reduction method of Barvinok's, and a call to a local nonlinear programming solver.

Short Bio:
Leo Liberti obtained his Ph.D. in Global Optimization at Imperial College London, held postdoctoral fellowships at Politecnico di Milano and Ecole Polytechnique in France, where he became a professor and vice-president of his department. After two years as a Research Staff Member at IBM Research in New York, he moved back to Europe as a Research Director at CNRS and part-time professor at Ecole Polytechnique. His main research interests are mixed-integer nonlinear programming and distance geometry.

The theory of rankings and voting is riddled with curious “paradoxes” related to non-transitivity. I will introduce a few new results on non-transitive dice games (joint work with Mike Paterson) and relate this to the area of Condorcet domains.
A Condorcet domain (CD) consists of a collection of linear orderings of a fixed finite set that satisfies a particular type of constraint for each 3-element subset.
One fundamental problem is to construct large CDs for a given finite set of size n. Already in the 1950s some constructions of CD domains were known, and these were then improved in the mid-1980s. Until recently only optimal values (9 and 20) for n=4 and n=5 were known. Recently, we showed (by heavy use of computer calculations) that the optimal values for n=6 and n=7 are 45 and 100 respectively (joint work with Akello-Egwel, Leedham-green, Litterrick, Markstrom). This result was already conjectured in the early 1990s. In the talk, I will present a new class of schemes for constructing large Condorcet domains that beat the 20+-year-old records. I will conclude with a list of open questions for further research.

In the early 70's Daykin conjectured that a family of subsets of
{1,2,...,n} which has the intersection property (i.e. any two subsets have
a non-empty intersection) and the non-union property (i.e the union of any
two subsets is not equal to {1,2,...,n}) contains at most 2^(n-2) members.
Different solutions of this conjecture were found within the next two of
three years by Schonheim, Seymour, (Daykin and Lovasz), and Hilton.
I will discuss various applications of the proofs and some
generalizations of the result.

For $n\geq 3$, an {$n$-spike is a matroid whose groundset can be partitioned into $n$ pairs called legs, such that the union of any two legs is both a circuit and a cocircuit. A spike may be extended by an element called a tip and/or coextended by an element called a cotip such that the union of the tip with each leg forms a triangle, and the union of the cotip with each leg forms a triad.
Spikes have arisen in many areas of matroid theory, sometimes notoriously, such as when Oxley, Vertigan & Whittle [1] used them to disprove Kahn's conjecture that the number of inequivalent representations of $3$-connected matroids over any finite field $GF(q)$ would be bounded above by some value $n(q)$.

I will aim this talk at a general combinatorial audience, spending some time on the widely-used geometrical representation of matroids and some of their properties. I will then present the characteristic polynomials, $\chi(M;\lambda)$, for the spike matroids (where the characteristic polynomial is a matroidal counterpart to the chromatic polynomial of a graph). I will also outline a proof that the zeros of these characteristic polynomials are bounded above by $\lambda=4$.
References
[1] Oxley, J., Vertigan, D., Whittle, G., On Inequivalent Representations of Matroids over Finite Fields. Journal of Combinatorial Theory, Series B 67, 325{343 (1996).

We prove the coefficientwise positivity for a class of multivariate formal power series that arise as fractional powers of determinants. More precisely, we show that the formal power series $1 - \det(I_n - tA_n)^{1/n}$ is coefficientwise nonnegative when $A_n$ is an $n \times n$ matrix built in a suitable way out of $n^2$ independent indeterminates.

This is joint work with Alex Scott.

A graph can be viewed, in many respects, as an analogue of an
algebraic curve. For example, there is a notion of "Jacobian" for
graphs.
More classically, graphs can be viewed as electrical networks.
I will discuss the interplay between these two points of view, as well
as some recent applications to problems in algebraic geometry.

In the 1960s and 1970s computer scientists attempted to classify the most prominent data structures in terms of the permutations they could sort. That is, how many of the n! permutations of length n could be sorted, within that data structure, to give as output the identity permutation? This study then gave rise to the broader study of questions such as "how many permutations avoid a given pattern of length 3, or 4 or 5?" We review the field, and present some solutions and conjectures for previously open problems. (Joint work with Andrew Conway and Andrew Elvey-Price).

We solve a generalised form of a conjecture of Kalai motivated by attempts to improve the bounds for Borsuk’s problem. The conjecture can be roughly understood as asking for an analogue of the Frankl-Rödl forbidden intersection theorem in which set intersections are vector-valued. We discover that the vector world is richer in surprising ways: in particular, Kalai’s conjecture is false, but we prove a corrected statement that is essentially best possible, and applies to a considerably more general setting. Our methods include the use of maximum entropy measures, VC-dimension, Dependent Random Choice and a new correlation inequality for product measures. Joint work with Peter Keevash.

In the general 'K-means problem', we are given n input points in a Euclidean space and seek to find k "center" points in the space so that the sum of the squared distances of each input point to its nearest center is minimized. While there have been several recent results for the special case in which the Euclidean space has some bounded dimension d, the best known approximation in arbitrary Euclidean spaces has remained (9+\epsilon) since 2002. In this talk I will present a new algorithm that achieves a 6.36-approximation for this problem, as well as an improved 2.64 approximation for the Euclidean k-median problem. The algorithm is based on a new Lagrangian multiplier preserving primal dual approach. This talk is based on joint work with Sara Ahmadian, Ashkan Norouzi-Fard, and Ola Svensson (to appear in FOCS '17).

The Graham-Pollak theorem asserts that, to decompose the complete graph on n vertices into complete bipartite subgraphs, we need at least n-1. What happens for hypergraphs: how many complete r-partite r-graphs are needed to decompose the complete r-graph? (Joint work with Luka Milicevic and Ta Sheng Tan.)

We consider the problem of covering a regular graph by a small number of vertex-disjoint paths. Magnan and Martin conjectured that every k-regular graph on n vertices can be partitioned into at most n / (k + 1) paths, a bound which is attained by a disjoint union of (k + 1)-cliques. We prove the conjecture in the case where k is linear in n and n is large.

This talk is based on joint work with Shoham Letzter.

We present a simple mechanism, which can be randomised, for constructing sparse 3-uniform hypergraphs with strong expansion properties. These hypergraphs are constructed using Cayley graphs over elementary abelian 2-groups and have vertex degree which is polylogarithmic in the number of vertices. Their expansion properties, which are derived from the underlying Cayley graphs, include analogues of vertex and edge expansion in graphs, rapid mixing of the random walk on the edges of the skeleton graph, uniform distribution of edges on large vertex subsets and the geometric overlap property.

We show that given any $r$-uniform hypergraph $F$, the trivially necessary divisibility conditions are sufficient to guarantee an edge-decomposition of any sufficiently large complete $r$-uniform hypergraph into edge-disjoint copies of $F$. The case when $F$ is complete corresponds to the existence of block designs, a problem going back to the 19th century, which was recently settled by Keevash. In particular, our argument provides a new proof of this result, which employs purely probabilistic and combinatorial methods. We also obtain several further generalizations.
This is joint work with Stefan Glock, Daniela K\"uhn and Deryk Osthus.

TBA

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By Ulm's theorem, countable reduced abelian p-groups are characterized, uniquely up to isomorphism, by their Ulm invariants. Given a sequence f of Ulm invariants, we provide a probabilistic construction of a countable abelian p-group G_f, having the set of natural numbers as its domain, with Ulm invariants ≤ f. We then show that with probability 1, G_f has precisely f as its sequence of Ulm invariants. This establishes the existence part of Ulm's theorem in a probabilistic way. We also develop new results for valuated abelian p-groups which are essential for our construction.

Joint work with Ruediger Goebel (Essen).

Covering designs are a generalisation of t-designs, where the requirement that any t-subset of points be contained in exactly λ blocks is replaced with the weaker requirement that they be contained in at least λ blocks. Covering arrays generalise orthogonal arrays in a similar manner.

In this talk, inspired by PJC's "generalised t-designs", I will present a common generalisation of covering designs and covering arrays, as well as some methods of constructing these new designs. In particular, I'll focus on the case t=2, where there is a strong relationship with graph theory, in the form of clique coverings.

If time permits, I may also talk about the "dual" problem of generalised packing designs.

In this talk we consider sets of vectors S of the vector space F_q^k with the property that every subset of S of size k is a basis.

The classical example of such a set is the following.

Example. (Normal rational curve) The set

 S = {(1,t,t²,...,t^k-1 | t∈F_q} ∪ {(0,,...,0,1)},

is a set of size q+1. It is easily shown that S has the required property by checking that the k×k Vandermonde matrix formed by k vectors of S has non-zero determinant.

For q even and k=3, one can add the vector (0,1,0) to S and obtain an example with q+2 vectors. For these parameters, such a set of q+2 vectors is called a hyperoval, and these have been studied extensively. There are many examples of hyperovals known which are not equivalent (up to change of basis and field automorphisms) to the example above. The only other known examples of size q+1 are an example of size 10 in F₉⁵, due to Glynn, and an example in F_2^h⁴ due to Hirschfeld.

The following conjecture exists in various areas of combinatorics. It is known as the main conjecture for maximum distance separable codes, the representability of the uniform matroid in matroid theory, the embeddability of the complete design in design theory, and Segre's arcs problem in finite geometry.

Conjecture. A set of vectors S of the vector space F_q^k, with k≤q–1, with the property that every subset of S of size k is a basis, has size at most q+1, unless q is even and k=3 or k=q–1, in which case it has size at most q+2.

I shall present a proof of the conjecture for q prime and discuss the non-prime case. I shall also explain how to then prove the following theorem, which is a generalisation of Segre's "oval is a conic" theorem in the case k=3.

Theorem. If p≥k, then a set S of q+1 vectors of the vector space F_q^k with the property that every subset of S of size k is a basis, is equivalent to a normal rational curve, where q=p^h.

Jacobson and Matthews introduced the most hopeful method known for efficiently generating uniformly distributed Latin squares. I will discuss how the same methods can be extended to generate uniformly distributed generalisations of Latin squares and lambda-factorisations of the complete graph at random.

The Ramsey Number, R(k), is defined as the minimum N such that every 2-coloring of the edges of K_N (the complete graph on N vertices) yields a monochromatic k-clique. For 60 years it is known that 2^k/2 < R(k) < 4^k, and it is a widely open problem to find significantly better bounds.

In this talk we consider a game theoretic variant of the Ramsey Numbers: Two players, called Maker and Breaker, alternately claim an edge of K_N. Maker's goal is to completely occupy a K_k and Breaker's goal is to prevent this. The game theoretic Ramsey Number R′(k) is defined as the minimum N such that Maker has a strategy to build a K_k in the game on K_N.

In contrast to the ordinary Ramsey Numbers, R′(k) has been determined precisely – a result of Beck. We will sketch a new, weaker result about R′(k) and use it to solve some related open problems.

Block character of a finite symmetric group is a  positive definite function which depends only on the number of cycles in  permutation. We describe the cone of block characters by
identifying its extreme rays, and find relations of the characters  to descent representations and the co invariant algebra of Sym_n. The decomposition of extreme block characters into the sum of characters of irreducible representations gives rise to certain limit shape theorems for random Young diagrams. We also study counterparts of the block characters for the infinite symmetric group Sym_∞, along with their connection to the Thoma characters of the infinite linear group GL_∞(q) over a Galois field.

Acyclic orientations of graphs have a number of applications, including a heuristic for graph colouring. We will discuss why we are interested in them and some investigations. Stanley showed that the number of acyclic orientations of a graph is, apart from sign, the evaluation of the chromatic polynomial at −1. There is a recurrence for the average number of acyclic orientations of a graph with a given number of vertices and edges; we would like to have further information on this distribution (for example, the variance). Given two acyclic orientations, it is possible to move from one to the other by reversing only the edges whose orientations differ, preserving acyclicity at each step. This gives rise to various Markov chains for choosing at random. The rate of convergence of these is an open problem.

The classical result of Erdős and Rényi shows that the random graph G(n,p) experiences sharp phase transition around p = 1/n – for any ε>0 and p = (1-ε)/n, all connected components of G(n,p) are typically of size O(log n), while for p = (1+ε)/n, with high probability there exists a connected component of size linear in n. We provide a very simple proof of this fundamental result; in fact, we prove that in the supercritical regime p = (1+ε)/n, the random graph G(n,p) contains typically a path of linear length. We also discuss applications of our technique to other random graph models and to positional games.

Joint work with M. Krivelelvich.

The chromatic polynomial P_X(q) of a graph X has the property that its value at a positive integer q is the number of proper colourings of the graph with q colours. However, the number it gives you may not be the number you want, for two reasons:

• it doesn't allow for the fact that different colourings may be equivalent under an automorphism of the graph;
• the order of the colours is significant, whereas you may want to count the partitions of the graph into independent sets.

I will present separate solutions for these two problems; one uses the orbital chromatic polynomial, while the other is a simple inclusion-exclusion argument. However, combining the two methods to count partitions into independent sets up to automorphisms is an unsolved problem.

Another feature of the chromatic polynomial is that substituting −1 for q counts the number of acyclic orientations of the graph. It turns out that substituting −1 for q in the orbital chromatic polynomial does not count orbits on acyclic orientations; but there is a "twisted" version of the orbital chromatic polynomial which does this job.

These things will be illustrated by means of the Petersen graph.

Given an undirected graph with costs on the edges and a positive integer k, consider the problem of finding a minimum cost spanning subgraph that is k-node-connected. We present a 6-approximation algorithm for this NP-complete problem, assuming that the number of nodes is at least k³(k−1)+k. This gives the first constant factor approximation for the problem.

For edge-connectivity variants, constant factor approximation can be achieved using the iterative rounding of the linear programming relaxation, a powerful technique developed by Jain. Whereas it has been long known that iterative rounding cannot yield a constant factor approximation for node-connectivity on arbitrary input graphs, we show that it does give a 2-approximation for a restricted class of instances. We apply a combinatorial preprocessing, based on the Frank–Tardos algorithm for k-outconnectivity, to transform any input into such an instance. This is a joint work with Joseph Cheriyan.

We observe that Ω(log n) chords must be added to an n-cycle to produce a pancyclic graph; we ask how much this must be increased in order that, given k with 3≤k≤n, there exists a cycle of each length ≥k which passes exactly kchords. We find that, for fixed k, the bound becomes Ω(n^1/k).

A finite deterministic automaton is synchronizing if there is a sequence of transitions which ends up at the same state no matter which state we start from.

There has been a lot of interest in the case where the transitions of the automaton consist of the generators of a permutation group G and a single non-permutation f: I will say that G synchronizes f if ⟨G,f⟩ is synchronizing.

It is known that a permutation group which synchronizes every non-permutation is primitive, but the converse is false. An important theorem of Rystsov asserts that a group of degree n is primitive if and only if it synchronizes every map of rank n−1; also, a primitive group synchronizes any map of rank 2. João Araújo has conjectured that a primitive group synchronizes every non-uniform map (one whose kernel classes do not all have the same size).

I will report on some progress on this conjecture that João and I made last month, including an improvement of Rystsov's theorem from n−1 to n−2.

Last term Peter Keevash and Fiachra Knox both gave talks on perfect matchings in hypergraphs. In this self-contained talk we look at the problem of establishing minimum degree conditions which force a hypergraph to contain a perfect matching.

Indeed, given positive integers k and r with k/2 ≤ r ≤ k−1, we give a minimum r-degree condition that ensures a perfect matching in a k-uniform hypergraph. This condition is best possible and improves on work of Pikhurko, who gave an asymptotically exact result. Our approach makes use of the absorbing method.

This is joint work with Yi Zhao (Georgia State University).

We show that for every ε>0 there exist δ>0$ and n₀∈N such that every 3-uniform hypergraph on n≥n₀ vertices with the property that every k-vertex subset, where k≥δn, induces at least (1/4+ε){k choose 3} edges, contains K₄⁻ as a subgraph, where K₄⁻ is the 3-uniform hypergraph on 4 vertices with 3 edges. This question was originally raised by Erdős and Sós. The constant 1/4 is the best possible.

This is a joint work with Roman Glebov and Dan Kral.

A maximal partial ovoid of a generalized quadrangle is a maximal set of points no two of which are collinear. The problem of determining the smallest size of a maximal partial ovoid in quadrangles has been extensively studied in the literature. In general, theoretical lower bounds on the size of a maximal partial ovoid in a quadrangle of order (s,t) are linear in s. I will discuss a simple probabilistic construction of a maximal partial ovoid of size at most s(log s)^α for a large class of quadrangles.

This is a rehearsal for a talk I will give at a meeting on Combinatorics and Physics in Cardiff the week after the end of term.

A combinatorial (or set-theoretic) solution of the Yang–Baxter equation is a set X (usually finite) and a function r:X²→X² satisfying the braiding condition on X³. With some other natural assumptions, combinatorial solutions give rise to abstract groups and permutation groups, related by homomorphism. The groups are soluble, and the derived length of the abstract group is one greater than that of the permutation group.

This is joint work with Tatiana Gateva-Ivanova, but I will be concentrating on the parts where I had some input.

In the early 1990's Laczkovich showed that, for any two sets A and B in
R^n with the same non-zero Lebesgue measure and with boundary of box
dimension less than n, there is a partition of A into finitely many
parts that can be translated by some vectors to form a partition of B.
In particular, this resolved the long-standing Tarki's circle squaring
problem.

We show that, in Laczkovich' theorem, one can additionally require
that all parts are Lebesgue measurable. This is a joint work with
András Máthé and Łukasz Grabowski.

What is the probability that the number of triangles in an Erdős–Rényi random graph exceeds its mean by a constant factor? In this talk, I will discuss some recent progress on this problem.

Already the order in the exponent of the tail probability was a long standing open problem until several years ago when it was solved by DeMarco and Kahn, and independently by Chatterjee. We now wish to determine the exponential rate of the tail probability. Thanks for the works of Chatterjee--Varadhan (dense setting) and Chatterjee--Dembo (sparse setting), this large deviations problem reduces to a natural variational problem. We solve this variational problem asymptotically, thereby determining the large deviation rate, which is valid at least for p > 1/n^c for some c > 0.

Based on joint work with Bhaswar Bhattacharya, Shirshendu Ganguly, and Eyal Lubetzky.

We show that for any positive integer r there exists an integer k and a k-colouring of the complete graph K_{2^{k} + 1} such that the colouring contains no monochromatic odd cycle of length less than r. This answers a question of Erdős and Graham. We use these colourings to give new lower bounds on the k-colour Ramsey number of the odd cycle and prove that, for all odd r and all k sufficiently large, there exists a constant c = c(r) > 0 such that the k-colour Ramsey number of C_{r} is at least (r-1)(2+c)^{k-1}.

This is joint work with Robert Johnson.

Network geometry aims at characterizing networks by means of geometrical tools and concepts. Network geometry is fundamental for systems as different as the brain or the quantum universe. After a broad overview of this field, recent works on network geometry will be presented. Here network geometry is modelled by means of simplicial complexes constructed not only by nodes and links but also by triangles and higher dimensional simplices. We will introduce Complex Quantum Network Manifolds (CQNM) describing the non-equilibrium evolution of growing simplicial complexes of dimension d. We show that in d=2 CQNM are homogeneous networks while for d >2 they are scale-free i.e. they are characterized by large inhomogeneities of degrees like most complex networks. From the self-organized evolution of CQNM quantum statistics emerge spontaneously. Here we define the generalized degrees associated with the faces of the d-dimensional CQNMs, and we show that the statistics of these generalized degrees can either follow Fermi-Dirac, Boltzmann or Bose-Einstein distributions depending on the dimension of the faces. A generalisation of these models called network geometry with flavor will be presented. Finally ensembles of simplicial complexes of dimension d will be characterized, in particular the configuration model of simplicial complexes and the canonical ensemble. The entropy of these ensembles will be given and the asymptotic expression for the number of simplicial complexes in the configuration model will be discussed.

A linearly constrained framework in d-dimensions is a d-dimensional bar-and-joint framework such that each vertex is allowed to move on a specific hyperplane. We denote linearly constrained frameworks by a triple (G, p, q) where G = (V, E) is a graph, p : V → R^d is the realisation map for the vertices, and q : V → R^d is the map that assigns unit vectors to the vertices that are normal to the associated hyperplanes. A linearly constrained framework is rigid if the only continuous motion of the vertices which satisfies the hyperplane constraints for the vertices and the length constraints for the edges, is the trivial motion which keeps each vertex fixed. We say that (G, p, q) is generic if (p, q) is algebraicly independent over the rationals. Streinu and Theran characterised generic rigidity of linearly constrained frameworks in R^2. In this talk we will give a characterisation for generic rigidity of linearly constrained frameworks in R^3. This is joint work with Bill Jackson.

Dirac's theorem states that any n-vertex graph with minimum degree at least n/2 contains a Hamilton cycle. Rödl, Ruciński and Szemerédi showed that asymptotically the same bound gives a tight Hamilton cycle in any k-uniform hypergraph, where in this case "minimum degree" is interpreted as the minimum codegree, i.e. the minimum over all (k-1)-sets of the number of ways to extend that set to an edge. The notion of a tight cycle can be generalised to an l-cycle for any l < k, and corresponding results for l-cycles were proved independently by Keevash, Kuhn, Mycroft and Osthus and by Han and Schacht, and extended to the full range of l by Kuhn, Mycroft and Osthus. However, l-cycles are essentially one-dimensional structures. A natural topological generalisation of Hamilton cycles in graphs to higher-dimensional structures is to ask for a spanning triangulation of a sphere in a 3-uniform hypergraph. We give an asymptotic Dirac-type result for this problem. Joint work with Agelos Georgakopoulos, Richard Montgomery and Bhargav Narayanan.

The area of mechanism design in economics is concerned with algorithms that produce good outcomes given inputs from self-interested individuals. One of the stars of mechanism design theory is the Vickrey-Clarke-Groves (VCG) mechanism, which makes it optimal for each individual to truthfully reveal its input. In practice the VCG mechanism is used with surprising rarity and the mechanisms we actually find are non-truthful. An example of this phenomenon are position auctions used by internet search engines to allocate space for advertisements displayed next to genuine search results. Here only non-truthful mechanisms have ever been used, and what in theory looks like a beginner's mistake was a huge commercial success. An obvious advantage of the non-truthful mechanisms is that they use simpler payments, and it has been known for some time why they do not produce chaos when participants decide strategically how to bid. We will see that the simplicity of payments also comes with a greater robustness to uncertainty on part of the search engines regarding the relative values of positions. This talk is based on joint work with Paul Dütting (LSE) and David C. Parkes (Harvard).

We say that a graph G is saturated with respect to some graph F if G doesn't contain any copies of F but adding any new edge to G creates some copy of F. The saturation number sat(F,n) is the minimum number of edges an F-saturated graph on n vertices can have. This forms an interesting counterpoint to the Turan number; the saturation number is in many ways less well-behaved. For example, Tuza conjectured that sat(F,n)/n must tend to a limit as n tends to infinity and this is still open. However, Pikhurko disproved a strengthening of Tuza's conjecture by finding a finite family of graphs, whose saturation number divided by n does not tend to a limit. We will prove a similar result for hypergraphs and discuss some variants.

The Tutte polynomial is one of the most important and well-known graph polynomials, and also features prominently in matroid theory. It is however not directly applicable to polymatroids, these being a natural generalisation of matroids. For instance, deletion-contraction properties do not hold. We construct a polynomial for polymatroids which behaves similarly to the Tutte polynomial of a matroid, and in fact contains the same information as the Tutte polynomial when we restrict to matroids. This is based on joint work with Alex Fink

A partition of a graph $G$ is equitable if, for any two parts $P_i$ and $P_j$ (possibly equal) and $x\in P_i$, the number $p_{ij}$ of neighbours of $x$ in $P_j$ depends only on $i$ and $j$ and not on $x$. It is easy to show that the spectrum of the matrix $(p_{ij})$ is contained in the spectrum of the graph.

Given a Latin square $L$ of order $n$, the corresponding Latin square graph has as vertex set the set of cells of the square, with two vertices joined if they lie in the same row or column or contain the same letter. Its eigenvalues are $3(n-1)$, $n-3$ and $-3$. We have proved that in an equitable partition of a Latin square graph whose matrix does not have $-3$ as an eigenvalue, each part is a union of rows, or of columns, or of letters, or of inflations of corner sets in Cayley tables of cyclic groups. I will also discuss some variants and generalisations.

This is joint work with Rosemary Bailey (QMUL/St Andrews) and Sergey Goryainov (Shanghai Jiao Tong University).

In the classical Private Information Retrieval (PIR) model, a user stores some data on a collection of servers and later wishes to retrieve a single bit of that data. This should be done without any individual server knowing which bit is being retrieved. The aim is to design a protocol that minimises the amount of communication between the user and servers. Variations of this model have been studied recently, motivated by applications in distributed storage. I will give a brief introduction to the area, and I will present some recent constructions and open problems. No knowledge of cryptography or distributed storage is required.

How many induced cycles can a graph on n vertices contain? For sufficiently large n, we determine the maximum number of induced cycles and the maximum number of even or odd induced cycles. We also characterize the graphs achieving this bound in each case. This answers a question of Tuza, and a conjecture of Chvátal and Tuza from 1988. Joint work with Alex Scott.

The subject of spatial voting considers how candidates compete for vote share in a society whose opinions can be expressed in some geometric opinion space. The most classical result here is the Median Voter Theorem which describes the case when opinions lie in a 1-dimensional interval. In higher dimensions the situation is much more complicated and there is no analogue of the Median Voter Theorem.

We investigate what can be said when the opinion space is the discrete hypercube (corresponding to d binary issues). This discrete model has been much less studied than the continuous ones and leads to some appealing problems in the combinatorics of the hypercube. We exhibit some intriguing behavior, results and open questions.

Joint Work with Nicholas Day

We prove a version of Tao's algebraic regularity lemma for asymptotic classes in the context of graphons. We apply it to study expander difference polynomials over fields with powers of Frobenius. This is joint work with Mirna Dzamonja (UEA).

We give a constant-factor approximation algorithm for the asymmetric traveling salesman problem. Our approximation guarantee is analyzed with respect to the standard LP relaxation, and thus our result confirms the conjectured constant integrality gap of that relaxation. Our techniques build upon the constant-factor approximation algorithm for the special case of node-weighted metrics. Specifically, we give a generic reduction to structured instances that resemble but are more general than those arising from node-weighted metrics. For those instances, we then solve Local-Connectivity ATSP, a problem known to be equivalent (in terms of constant-factor approximation) to the asymmetric traveling salesman problem. This is joint work with Ola Svensson and Jakub Tarnawski.

The (p,q)-Maker-Breaker-percolation game is played by two players, Maker and Breaker, who take turns claiming edges from the two-dimensional integer lattice.  On each of their turns Maker claims p different unclaimed edges, while on each of their turns Breaker claims q different unclaimed edges.  Maker's aim is to build arbitrarily long paths from the origin of the lattice, while Breaker's aim is to prevent this from happening.  We say that Maker wins the game if they can guarantee that, among the set of unclaimed edges and edges that Maker has claimed, there is always a path from the origin escaping to infinity.

In this talk we will discuss such Maker-Breaker-percolation games and give a number of results for when Maker or Breaker can win.  To do this we will look at and discuss certain Maker-Breaker crossing games.  These games are similar to the above percolation games, except that they are played on finite two dimensional grids and Maker's aim is to create a path of edges that crosses from the left side of the grid to the right side.  We will give a number of results for such Maker-Breaker crossing games, show how they relate back to the original percolation games, and discuss how they are related to the combinatorial games of Hex, Bridg-it and the Shannon switching game.

This talk is based on joint work with Victor Falgas-Ravry.

The topic of this talk is the computational complexity of the homomorphism problem between two relational structures, also known as the constraint satisfaction problem (CSP). We briefly discuss the known classifications of CSPs parametrised by the source or target structures. We then discuss a classification of general-valued CSPs parametrised by the source structures. Based on joint work with Clement Carbonnel and Miguel Romero.

The distribution of the degree sequence of a random graph has been studied since the original papers on random graphs by Erdos and Renyi.  About 25 years ago, results on asymptotic enumeration of graphs with given degree sequence prompted McKay and the speaker to conjecture a simple model for the degree sequence of a random graph. I will discuss the recent resolution of this conjecture, which was joint work with Anita Liebenau, and related results.

By the well known theorem of Sperner, the minimum number of chains the Boolean lattice (2[n], \subset) can be partitioned into is \binom{n}{\floor{n/2}}. There are many ways to construct such a chain partition, however, most of these partitions contain long and short chains at the same time. A conjecture of Furedi asks if there exists a partition of 2[n] into \binom{n}{\floor{n/2}} chains such that the difference between the size of any two chains is at most 1.

While this conjecture is still open, we managed to construct a chain partition, where the size of every chain is \Theta(\sqrt{n}). More interestingly, it turns out that these chain partitions have applications in certain extremal set theory problems. We outline our results concerning the conjecture of Furedi and discuss the possible applications in more detail.

A clique in a graph is a regular clique if every vertex not in the clique is adjacent to the same number of vertices inside the clique. In the early 1980's, Neumaier studied the existence of regular cliques in edge-regular graphs. The examples of edge-regular graphs with regular cliques that Neumaier knew of were all strongly regular graphs, so he asked if there exist edge-regular, non-strongly regular graphs containing regular cliques. With this as motivation, we define a Neumaier graph as an edge-regular, non-strongly regular graph containing a regular clique. We will discuss the answer to Neumaier's question, and then focus on the determination of the smallest Neumaier graph. No prior knowledge of edge-regular or strongly regular graphs is assumed.

I have discovered a family of finite posets which arise in representation theory. Remarkably, they can be defined in six quite different but equivalent ways. I will give all six definitions and some basic properties of these posets. Only very elementary theory of posets will be assumed.

The Ramsey number $r(C_{\ell},K_n)$ is the smallest natural number $N$ such that every red/blue edge-colouring of a clique of order $N$ contains a red cycle of length $\ell$ or a blue clique of order $n$. In 1978, Erd\H{o}s, Faudree, Rousseau and Schelp conjectured that $r(C_{\ell},K_n) = (\ell-1)(n-1)+1$ for $\ell \geq n\geq 3$ provided $(\ell,n) \neq (3,3)$. In this talk I will discuss a recent proof of this conjecture for large $\ell$, and a strong form of a conjecture due to Nikiforov, showing that $r(C_{\ell},K_n) = (\ell-1)(n-1)+1$ provided $\ell \geq \frac {C\log n}{\log \log n}$, for some absolute constant $C>0$. Up to the value of $C$ this is tight, and answers two further questions of Erd\H{o}s et al. up to multiplicative constants.

Joint work with Peter Keevash and Jozef Skokan.

The deck of a graph G is given by the multiset of (unlabelled) sub-graphs \{G-v:v\inV(G)\}. The subgraphs G-v are referred to as the cards of G. Brown and Fenner recently showed that, for n\geq29, the number of edges of a graph G can be computed from any deck missing 2 cards. We will show that, for sufficiently large n, the number of edges can be computed from any deck missing at most 1/20 \sqrt{n} cards. This result is joint work with Alex Scott and Carla Groenland.

We show that in every two-colouring of the edges of the complete graph $K_N$ there is a monochromatic $K_k$ which can be extended in at least $(1 + o_k(1))2^{-k}N$ ways to a monochromatic $K_{k+1}$. This result is asymptotically best possible, as may be seen by considering a random colouring. Equivalently, defining the book $B_n^{(k)}$ to be the graph consisting of $n$ copies of $K_{k+1}$ all sharing a common $K_k$, we show that the Ramsey number $r(B_n^{(k)}) = 2^k n + o_k(n)$. In this form, our result answers a question of Erd\H{o}s, Faudree, Rousseau and Schelp and establishes an asymptotic version of a conjecture of Thomason.

The first reference to the rigidity of frameworksin the mathematical literature occurs in a problem posed by Euler in 1776. Consider a polyhedron P in 3-space. We view P as a `panel-and-hinge framework' in which the faces are 2-dimensional panels and the edges are 1-dimensional hinges. The panels are free to move continuously in 3-space, subject to the constraints that the shapes of the panels and the adjacencies between themare preserved, and that the relative motion between pairs of adjacent panels is a rotation about their common hinge. The polyhedron P is rigid if every such motion results in a polyhedron which is congruent to P.

Euler conjectured that every polyhedron is rigid. This conjecture was verified for the case when P is convex by Cauchy in 1813. Gluck showed in 1975 that it is true when P is `generic' i.e. there are no algebraic dependencies between the coordinates of the vertices of P. Connelly finally disproved the conjecture in 1982 by constructing a polyhedron which is not rigid.

 I will describe results and open problems concerning the rigidity of various other types of frameworks. I will be mostly concerned with the generic case for which the problem of characterizing rigidity usually reduces to pure graph theory.

The Tur\'an function $ex(n,F)$ denotes the maximal number of edges in an $F$-free graph on $n$ vertices. However once the number of edges in a graph on $n$ vertices exceeds $ex(n,F)$, many copies of $F$ appear. We study the function $h_F(n,q)$, the minimal number of copies of $F$ in a graph on $n$ vertices with $ex(n,F)+q$ edges. The value of $h_F(n,q)$ has been extensively studied when $F$ is colour critical. We consider a non-colour-critical graph, namely the bowtie and establish bounds on $h_F(n,q)$ when $q=o(n^2)$.

This is joint work with Mihyun Kang and Oleg Pikhurko.

By Cayley's theorem, the number of trees on the vertex set $\{1,\ldots,n\}$ is

$n^{n-2}$. In such a tree, we can regard each edge as a transposition. If

we multiply together these transpositions (in any order), we obtain an

$n$-cycle. (The simple proof resembles the argument to show the number of

breaks needed to convert a chocolate bar into separate squares.) Since there

are $(n-1)!$ orderings of the edges, and $(n-1)!$ cycles, all conjugate, one

might first think that a given tree produces each cycle exactly once. However,

this is false; the only trees for which it holds are stars.

So one can ask about the number and distribution of cycles produced by a given

tree, and (the inverse problem) the number of trees which give rise to a given

cycle. We have some results on these questions. An interesting feature is the

appearance of some famous combinatorial sequences such as the Euler numbers

and the Fuss--Catalan numbers.

This is joint work with St Andrews undergraduate Liam Stott, done during his

summer research internship last summer.

No Seminar.

A perfect packing of a family $(G_i)_{i\in[m]}$ of graphs in a host graph $H$ is a colouring of the edges of $H$ with colours $[m]$ such that the edges with colour $i$ form a copy of $G_i$ for each $i\in[m]$. We show that for large $n$ we can perfectly pack any family $(G_i)_{i\in[m]}$ of $d$-degenerate graphs with maximum degree at most $cn/\log n$ into~$K_n$ if $\sum e(G_i)=n(n-1)/2$ and if for $i\ge m-\eta n$ we have  $v(G_i)\le(1-\eta)n$ and that~$G_i$ has at least $\eta n$ leaves.

This is joint work with Peter Allen, Dennis Clemens, Anusch Taraz.

Valuations are a classical topic in convex geometry. The volume plays an
important role in many structural results, such as Hadwiger’s famous
characterization of continuous, rigid-motion invariant valuations on
convex bodies. Valuations whose domain is restricted to lattice
polytopes are less well-studied. The Betke-Kneser Theorem establishes a
fascinating discrete analog of Hadwiger’s Theorem for lattice-invariant
valuations on lattice polytopes in which the number of lattice points —
the discrete volume — plays a fundamental role. In this talk, we explore
striking parallels, analogies and also differences between the world of
valuations on convex bodies and those on lattice polytopes with a focus
on positivity questions and links to Ehrhart theory.

Given a smooth curve defined over a valued field, it is a difficult problem to compute the Berkovich skeleton of the curve. In theory, one can find a semistable model for the curve and then find the dual graph of the special fiber, and this will give the skeleton. In practice, these procedures are not algorithmic and finding the model can become difficult. In this talk, we present a method for doing this in the case of superelliptic curves y^n=f(x). The solution is combinatorial and involves studying the covering from the curve to $\mathbb{P}^1$, and recovering data about the Berkovich skeleton from the tropicalization of $\mathbb{P}^1$ together with the marked ramification points. Throughout the talk we will study many examples in order to get a feel for the difficulties of this problem and how the procedure is carried out.

We give an FPTAS and an efficient sampling algorithm for the high-fugacity hard-core model on bounded-degree bipartite expander graphs and the low-temperature ferromagnetic Potts model on bounded-degree expander graphs.  The results apply, for example, to random (bipartite) $\Delta$-regular graphs, for which no efficient algorithms were known for these problems (with the exception of the Ising model) in the non-uniqueness regime of the infinite $\Delta$-regular tree. Joint work with Peter Keevash and Will Perkins.

A spatial random permutation (SRP) is a model of random permutation that is biased towards the identity in some underlying geometry; well-known examples include the Mallows permutation and Feynman's representation of the Bose gas. SRPs are expected to exhibit certain universal phenomena, such as crossover behaviour in the mean displacement ("band structure") and the emergence of macroscopic cycles when the band exceeds a critical width. In this talk we study the band structure of a Boltzmann model of SRP on a lattice in which the energy is equal to the total displacement. Our analysis rests on a surprising connection between random permutations and Gaussian processes, via the matrix permanent; even though this connection is well-known in other contexts, its application to random permutations is to the best of our knowledge novel. Joint work with Yan Fyodorov (King's College London).

Matching fields are a collection of matchings on bipartite node sets satisfying an "edge exchange axiom". They were first considered by Sturmfels and Zelevinsky in the 90s as a combinatorial tool to study the algebraic variety of degenerate matrices. More recently, they have gained attention via their connection to tropical linear algebra, specifically to encode "tropical linear dependence". We will consider these objects and the natural ways in which they arise via the combinatorics of matrices. We shall also introduce some additional collections of bipartite graphs arising from tropical linear algebra and, in the process, answer two conjectures of Sturmfels and Zelevinsky on the structure of degenerate matrices.

The origin of the longest monotone sequence selection problem lies in the infamous Ulam's problem of the longest possible increasing sequence in the random permutation of numbers. Although the latter problem was settled to a somewhat satisfactory degree, the online setup requires a completely different approach. Introduced first by Samuels and Steele, it has led to the emergence of numerous variations on the original problem. In this talk we focus on the classical problem, its dual counterpart, and a poissonised (continuous) version of the classical problem. The latest results on asymptotics of the value functions are discussed, together with the properties of several selection policies.

Tropical geometry is a relatively recent field that studies 'polyhedral
shadows' of algebraic varieties. In the tropical world there are many
objects and theorems which are analogous to those in algebraic geometry.
In this talk we will look specifically at tropical linear spaces which are
parametrized by valuated matroids. We will discuss the Dressian, which is
the space of all valuated matroids, and formulate some open questions. We
will focus on a particular class of valuated matroids that come from
matrices with tropical entries. These are the valuated analog of
transversal matroids, which are matroids that arise from bipartite graphs.
This is based on joint work with Alex Fink and, separately, with Marta
Panizzut and Benjamin Schröter.

In a recent breakthrough paper, Anari, Liu, Oveis Gharan and Vinzant – building on work of Kaufman and Oppenheim – settled a long-standing conjecture about the edge expansion of the bases-exchange graph of an arbitrary matroid.  That at least is the headline result, but many other interesting things flow out of their work.  I’ll spend the first half of the talk explaining the terms appearing in the title, and then go on to hint at some of the ideas involved and describe some of the algorithmic consequences. The talk will be a toric-ideal-free zone.

Tropical ideals are combinatorial objects introduced with the aim of giving tropical geometry a solid algebraic foundation. They can be thought of as discrete analogues of polynomial ideals in which any bounded-degree part is “matroidal”. I will introduce and motivate the notion of tropical ideals, and I will discuss work studying some of their main properties and their possible associated varieties.

In each round of the Namer-Claimer game (i) Namer names a forbidden distance d, then (ii) Claimer claims a subset of [n] not containing two points at distance d. How quickly can Claimer claim subsets covering [n] if Namer is trying to slow them down?  

There will be an answer, a surprising connection to Ramsey theory and several open problems.

For a graph H, the extremal number ex(n,H) is defined to be the maximal number of edges in an H-free graph on n vertices. For bipartite graphs H, determining the order of magnitude of ex(n,H) is notoriously difficult. In this talk I present recent progress on this problem.

The k-subdivision of a graph F is obtained by replacing the edges of F with internally vertex-disjoint paths of length k+1. Most of our results concern the extremal number of various subdivided graphs, especially the subdivisions of the complete graph and the complete bipartite graph.

Partially joint work with David Conlon and Joonkyung Lee.

In 1981 Jackson showed that the diregular bipartite tournament (a balanced complete bipartite graph whose edges are oriented so that every vertex has the same in- and outdegree) contains a Hamilton cycle, and conjectured that in fact the edge set of it can be partitioned into Hamilton cycles. We prove an approximate version of this conjecture: For every c>1/2 and ε>0 there exists n_0 such that every cn-regular bipartite digraph on 2n>n_0 vertices contains (1-ε)cn edge-disjoint Hamilton cycles. This is joint work with Anita Liebenau (UNSW Sydney).

The famous Erdős-Rademacher problem asks for the smallest number of $r$-cliques in a graph with the given number of vertices and edges. Despite decades of active attempts, the asymptotic value of this extremal function for all $r$ was determined only recently, by Reiher. In this talk, we describe the asymptotic structure of all almost extremal graphs. This task for $r=3$ was previously accomplished by Pikhurko and Razborov. This is joint work with Jaehoon Kim, Hong Liu, and Oleg Pikhurko.

A beautiful coupling argument of McDiarmid shows that, given any
orientated n-vertex cycle C, a copy of C almost surely appears in D(n,p)
if p=(log n+log log n+omega(1))/n. This is not always optimal—as Frieze
had shown, for a directed Hamilton cycle p=(log n+omega(1))/n is sufficient.

This talk will address the following questions, by combining McDiarmid’s
coupling with constructive techniques:

- Is p=(log n+omega(1))/n sufficient for the appearance of any oriented
n-vertex cycle?

- When is it likely that a copy of all such cycles can be found
simultaneously?

Hierarchical clustering is a recursive partitioning of a dataset into clusters at an increasingly finer granularity. Motivated by the fact that most work on hierarchical clustering was based on providing algorithms, rather than optimizing a specific objective, Dasgupta framed similarity-based hierarchical clustering as a combinatorial optimization problem, where a “good” hierarchical clustering is one that minimizes a particular cost function. He showed that this cost function has certain desirable properties: To achieve optimal cost, disconnected components (namely, dissimilar elements) must be separated at higher levels of the hierarchy, and when the similarity between data elements is identical, all clusterings achieve the same cost. We take an axiomatic approach to defining “good” objective functions for both similarity- and dissimilaritybased hierarchical clustering. We characterize a set of admissible objective functions having the property that when the input admits a “natural” ground-truth hierarchical clustering, the ground-truth clustering has an optimal value. We show that this set includes the objective function introduced by Dasgupta. Equipped with a suitable objective function, we analyze the performance of practical algorithms, as well as develop better and faster algorithms for hierarchical clustering. We also initiate a beyond worst-case analysis of the complexity of the problem and design algorithms for this scenario.

In 1989, Rota made the following conjecture. Given n bases B1, ..., Bn in an n-dimensional vector space V, one can always find n disjoint bases of V, each containing exactly one element from each Bi (we call such bases transversal bases). Rota’s basis conjecture remains open despite its apparent simplicity. In this talk, we will discuss how to find  (0.5 - o(1))n disjoint transversal bases, improving the previously best known bound of n/log n. This is joint work with Bucic, Kwan, and Sudakov.

In this talk I present a general and versatile algorithmic framework for exhaustively generating a large variety of different combinatorial objects, based on encoding them as permutations, which provides a unified view on many known results and allows us to prove many new ones. In particular, we obtain the following four classical Gray codes as special cases: the Steinhaus-Johnson-Trotter algorithm to generate all permutations of an n-element set by adjacent transpositions; the binary reflected Gray code to generate all n-bit strings by flipping a single bit in each step; the Gray code for generating all n-vertex binary trees by rotations due to Lucas, van Baronaigien, and Ruskey; the Gray code for generating all partitions of an n-element ground set by element exchanges due to Kaye.

The first main application of our framework is the generation of pattern-avoiding permutations, yielding new Gray codes for different families of permutations that are characterized by the avoidance of certain classical patterns, (bi)vincular patterns, barred patterns, mesh patterns, monotone and geometric grid classes, and many others. We also obtain new Gray codes for all the combinatorial objects that are in bijection to these permutations, in particular for five different types of geometric rectangulations, also known as floorplans, which are divisions of a square into n rectangles subject to different restrictions.

The second main application of our framework are lattice congruences of the weak order on the symmetric group S_n. Recently, Pilaud and Santos realized all those lattice congruences as (n-1)-dimensional polytopes, called quotientopes, which generalize hypercubes, associahedra, permutahedra etc. Our algorithm generates the equivalence classes of each of those lattice congruences, by producing a Hamilton path on the skeleton of the corresponding quotientope, yielding a constructive proof that each of these highly symmetric graphs is Hamiltonian. We thus also obtain a provable notion of optimality for the Gray codes obtained from our framework: They translate into walks along the edges of a polytope.

This is joint work with Liz Hartung, Hung P. Hoang, and Aaron Williams (SODA 2020).

A family of permutations is said to be 't-intersecting' if any two permutations in the family agree on at least t points. It is said to be (t-1)-intersection-free if no two permutations in the family agree on exactly t-1 points. Deza and Frankl conjectured in 1977 that a t-intersecting family of permutations in S_n can be no larger than a coset of the stabiliser of t points, provided n is large enough depending on t; this was proved by the speaker and independently by Friedgut and Pilpel in 2008. We give a new proof of a stronger statement: namely, that a (t-1)-intersection-free family of permutations in S_n can be no larger than a coset of the stabiliser of t points, provided n is large enough depending on t. This can be seen as an analogue for permutations of seminal results of Frankl and Furedi on families of k-element sets. Our proof is partly algebraic and partly combinatorial; it is more 'robust' than the original proofs of the Deza-Frankl conjecture, using a combinatorial 'quasirandomness' argument to avoid many of the algebraic difficulties of the original proofs. Its robustness allows easier generalisation to various other permutation groups. Based on joint work with Noam Lifshitz (Hebrew University of Jerusalem).

We study the Lonely Runner Conjecture, conceived by Jörg M. Wills in the 1960's: Given positive integers n_1, n_2, ..., n_k, there exists a positive real number t such that for all 1 ≤ j ≤ k the distance of tn_j to the nearest integer is at least 1/(k+1). Continuing a view-obstruction approach by Cusick and recent work by Henze and Malikiosis, our goal is to promote a polyhedral ansatz to the Lonely Runner Conjecture. Our results include geometric proofs of some folklore results that are only implicit in the existing literature, a new family of affirmative instances defined by the parities of the speeds, and geometrically motivated conjectures whose resolution would shed further light on the Lonely Runner Conjecture.

This is joint work with Serkan Hosten (SF State) and Matthias Schymura (Lausanne).

In this work we consider a generalisation of Kelly’s conjecture which is due Alspach, Mason, and Pullman from 1976. Kelly’s conjecture states that every regular tournament has an edge decomposition into Hamilton cycles, and this was proved by Kuehn and Osthus for large tournaments. The conjecture of Alspach, Mason, and Pullman concerns general tournaments and asks for the minimum number of paths needed in an edge decomposition of each tournament into paths.  There is a natural lower bound for this number in terms of the degree sequence of the tournament and they conjecture this bound is correct for tournaments of even order. Almost all cases of the conjecture are open and we prove many of them.

This is joint work with Allan Lo (Birmingham), Viresh Patel (UVA) and John Talbot (UCL).

We present a new technique for use in switching-based random
generation of graphs with given degrees. For graphs with m edges and
maximum degree D=O(m^4), the ``best'' existing uniform sampler, by McKay
and Wormald, runs in time O(m^2D^2). Our new one runs in time O(m),
which is effectively optimal. For d-regular graphs with d^2=o(n), the
best existing ones run in time O(nd^3). This is now improved  to
O(nd+d^4). Similar results are obtained for generating random
contingency tables with given marginals (equivalently, bipartite
multigraphs with given degree sequence) in the sparse case. This is
joint work with Andrii Arman and Jane Gao.

This is an informal talk, aimed at non-specialists.

We study the asymptotics of the number of standard Young tableaux of large skew diagrams.  We present new bounds and discuss how they compare with the existing general bounds.  I will also make a connection with the 1/3-2/3 conjecture for linear extension of posets.  Our approach is based on a recent Naruse's hook-length formula which I will also explain, and is related to weighted lozenge tilings.  The talk assumes no prior knowledge of the subject.

Given a solution to a problem, we can try and apply a series of elementary operations to it, making sure to remain in the solution space at every step. What kind of solutions can we reach this way? How fast? This is motivated by a variety of applications, from statistical physics to real-life scenarios, including enumeration and sampling. In this talk, we will discuss various positive and negative results, in the special case of graph colouring.

We study the problem of impartial selection, a topic that lies at the intersection of computational social choice and mechanism design. The goal is to select the most popular individual among a set of community members. The input can be modelled as a directed graph, where each node represents an individual, and a directed edge indicates nomination or approval of a community member to another. An impartial mechanism is robust to potential selfish behaviour of the individuals and provides appropriate incentives to voters to report their true preferences by ensuring that the chance of a node to become a winner does not depend on its outgoing edges. The goal is to design impartial mechanisms that select a node with an in-degree that is as close as possible to the highest in-degree.

Recent progress has identified impartial selection rules with optimal approximation ratios. It was noted though that worst-case instances are graphs with few vertices. Motivated by this fact, we propose the study of additive approximation, the difference between the highest number of nominations and the number of nominations of the selected member, as an alternative measure of quality. We present randomized impartial selection mechanisms with additive approximation guarantees of o(n), where n is the number of nodes in the input graph.

We furthermore demonstrate that prior information on voters' preferences can be useful in the design of simple (deterministic) impartial selection mechanisms with good additive approximation guarantees. In this direction, we consider different models of prior information and analyze the performance of a natural selection mechanism that we call approval voting with default (AVD).

The talk is based on joint work with George Christodoulou and Nicos Protopapas. No advanced mathematical background (besides basic knowledge on graphs and probability) is required to follow the talk.

Framework rigidity is concerned with questions of the following form: what can we learn about an unknown set of n points p_1, …, p_n in d-dimensional space from the distances between a subset of the pairs {p_i, p_j}, with i and j known?  Global rigidity of a framework (G, p) is then the question: can we recover p, up to an unknown isometry?

I will introduce and discuss variations of global rigidity problems when we are given lengths, but not the combinatorics of the measurement process that generated them. Maybe surprisingly, we can get positive results, under a suitable genericity hypotheses on the unknown configuration p.

The study of triangle-free graphs has a long tradition yet continues to produce novel insights. Classic results on the independence or chromatic number of triangle-free results may be viewed as leveraging local structural information -- that is, the absence of edges in any neighbourhood -- to produce global structure. In this talk, we mainly consider the obvious generalisation where instead of absence, some bound on the density of edges in any neighbourhood is assumed. This perspective has led to important general results, for example, by Alon, Krivelevich and Sudakov (1999), and by Molloy and Reed (1997). We discuss new probabilistic methods to handle this and related problems. This covers some joint works with Ewan Davies, Rémi de Joannis de Verclos, Eoin Hurley, François Pirot, Jean-Sébastien Sereni.

The concept of reachability sets (i.e. the set of vertices which can be reached from a given starting point by travelling along edges) is central to many network-based processes, including the dissemination of information or the spread of disease through a network. In many cases - for example, considering the spread of a biological disease, a computer virus, or fake news - it might be desirable to reduce the number of vertices reachable from any given starting vertex.  Moreover, in most settings, time plays a crucial role: each contact between individuals, represented by an edge, will only occur at certain time(s), when the corresponding edge is "active". The relative timing of edges is clearly crucial in determining the reachability set of any vertex in the graph.

In this talk, I will address the problems of reducing the maximum reachability of any vertex in a given temporal network by two different means:
(1) we can remove a limited number of edges from the graph, or
(2) every edge is retained, but we can change the relative order in which different edges are active (perhaps subject to some additional constraints).
Mostly, we find that these problems are computationally intractable even when very strong restrictions are placed on the input, but we identify a small number of special cases that admit polynomial-time or FPT algorithms, as well as giving some more general approximation algorithms.

Everything in this talk is based on joint work with Jessica Enright (University of Glasgow); I will also mention some joint results with George B. Mertzios (University of Durham), and Viktor Zamaraev (University of Liverpool) and Fiona Skerman (Uppsala University).

How many seats should be allocated to each political party in an election? This question has a long and rich history, and plays a fundamental role in order to ensure appropriate levels of representation in the parliaments. We consider this question in the presence of types (e.g. men and women), where apart from ensuring proportionality at the party level we also have to find an allocation for the types under parity constraints. We consider two different approaches: one of them, following a greedy/local search paradigm, corresponds to a mechanism recently approved in the Chilean parliament (March 2020) to find the representatives of the constitutional assembly with the goal of designing a new political constitution for Chile (election to happen in April 2021); the other one is based on the idea of integral biproportionality, introduced by Balinski and Demange in the late 80’s. We analyze both mechanisms from an axiomatic and quantitative point of view. We also provide new results regarding integral biproportional solutions and the fair share, a.k.a matrix scaling solution, which represents the ideal (but not necessarily implementable) fractional biproportional solution.

Many problems in combinatorial optimisation, data science, and machine learning can be cast as finding the optimum value of a submodular objective function subject to some combinatorial constraint. Although this problem is NP-hard even for a simple cardinality constraint, there are several classical algorithms that deliver constant-factor approximation guarantees for various constraints. Here, we consider what can be done in the following “streaming" setting. Suppose that a problem instance (i.e. the domain of some submodular function we seek to optimise) is too large to efficiently store in random access memory, and instead arrives as a stream of data elements (either once or multiple times). What can be done if we are allowed to store only a comparatively small number of these elements in the ground set? In this talk, I will discuss some recent join work with Theophile Thiery, that shows that by performing a constant number of passes through the data stream, we can obtain improved approximation guarantees nearly matching that of well-known local search algorithms for the standard, offline setting. Our techniques apply to monotone and non-monotone submodular maximisation under matroid constraints and also p-matchoid constraint, which generalise matching, set-packing, and matroid intersection problems.

Solving a conjeture of Bollobás from the 1970's, Komlós, Sárközy and Szemerédi proved in 1995 that every graph of minimum degree \(n/2 + o(n)\) contains every \(n\)-vertex tree of bounded maximum degree as a subgraph. We show an extension of this theorem to \(k\)-uniform hypergraphs. The trees in our theorem are tight k-trees. Our proof relies on a decomposition of tight k-trees, weak hypergraph regularity, and absorption. We will also give an overview of related questions for graphs and hypergraphs.
This is joint work with Matías Pavez-Signé and Nicolás Sanhueza-Matamala

How long does it take for a pandemic to stop spreading? In this talk, we consider this question in the bootstrap percolation setting.
Graph-bootstrap percolation, also known as weak saturation, was introduced by Bollobás in 1968. In this process, we start with initial "infected" set of edges E_0, and we infect new edges according to a predetermined rule. Given a graph H and a set of previously infected edges E_t ⊆ E(K_n), we infect a non-infected edge e if it completes a new copy of H in G=([n] , E_t ∪ {e}). A question raised by Bollobás asks for the maximum time the process can run before it stabilizes. Bollobás, Przykucki, Riordan, and Sahasrabudhe considered this problem for the most natural case where H=K_r. They answered the question for r ≤ 4 and gave a non-trivial lower bound for every r ≥ 5. They also conjectured that the maximal running time is o(n^2) for every integer r. We disprove their conjecture for every r ≥ 6 and we give a better lower bound for the case r=5; in the proof we use the Behrend construction. This is a joint work with József Balogh, Alexey Pokrovskiy, and Tibor Szabó.

A diregular bipartite tournament is a balanced complete bipartite graph whose edges are oriented so that every vertex has the same in- and outdegree. In 1981, Jackson showed that a diregular bipartite tournament contains a Hamilton cycle, and conjectured that in fact the edge set of it can be partitioned into Hamilton cycles. We prove an approximate version of this conjecture: for every epsilon>0 there exists n₀ such that every diregular bipartite tournament on 2n>n₀  vertices contains a collection of (1/2-epsilon)n cycles of length at least (2-epsilon)n. Increasing the degree by a small proportion allows us to prove the existence of many Hamilton cycles: for every c>1/2 and epsilon>0 there exists n₀ such that every cn-regular bipartite digraph on 2n>n₀ vertices contains (1-epsilon)cn$ edge-disjoint Hamilton cycles.

Joint work with Yani Pehova.

Similarly to the way that the digraph representing a permutation can be uniquely decomposed into cycles, the digraph representing an injection between two finite sets can be uniquely decomposed into cycles and paths. The component structure of a random injection undergoes a phase transition between cycles dominating and paths dominating as certain parameters change.

The Mullineux map is a function on partitions which arises in the representation theory of the symmetric group.  The Mullineux problem is to describe this function purely combinatorially.  I will describe the history of this problem and explain the known combinatorial solutions, and then give a new solution based on crystals and regularisation.

An (n,s,q)-graph is an n-vertex multigraph in which every s-set of vertices spans at most q edges. Turan-type questions on the maximum of the sum of the edge multiplicities in such multigraphs have been studied since the 1990s. More recently, Mubayi and Terry posed the problem of determining the maximum of the product of the edge multiplicities in

This is joint work with Edin Husić, Georg Loho and László Végh.

In the game of Bridg-it, two players – let us call them Alice and Bob – play in alternating turns by placing bridges on a rectangular board. On each of their turns, Alice adds a red bridge while Bob adds a blue bridge.  Alice's aim is to build a connected path of red bridges from the left-hand side of the board to the right-hand side, while Bob for his part tries to hinder her by assembling a connected path of blue bridges from the top side of the board to the bottom side.

Bridg-it was introduced by David Gale in the 1950s, and was instantly and completely solved: for each board we know both the identity of the winning player under optimal play and an explicit winning strategy. But what happens if the players were allowed to place more than one bridge on each of their turns – say, for example that Alice places two bridges and Bob three? Such variants of the game turn out to be considerably more difficult to analyse than the original Bridg-it. In this talk, I will describe some special cases where one can find explicit winning strategies (as well as some elementary cases that remain stubbornly open!), and how these are connected to percolation-inspired Maker-Breaker games.

Joint work with A. Nicholas Day

The semi-random graph process can be thought of as a one player game. Starting with an empty graph on n vertices, in each round a random vertex u is presented to the player, who chooses a vertex v and adds the edge uv to the graph. Given a graph property, the objective of the player is to force the graph to satisfy this property in as few rounds as possible.

We will consider the property of constructing a fixed graph G as a subgraph of the semi-random graph. Ben-Eliezer, Gishboliner, Hefetz and Krivelevich proved that the player can asymptotically almost surely construct G given n^{1 – 1/d}w rounds, where w is any function tending to infinity with n and d is the degeneracy of the graph G. We have proved a matching lower bound. I will talk about this result, and also discuss a generalisation of our approach to semi-random hypergraphs. I will finish with some open questions.

This is joint work with Trent Marbach, Pawel Pralat and Andrzej Rucinski.

I will provide an introductory talk to hypergraph varieties, focusing on the combinatorial aspect. I will not assume any prior knowledge of algebraic, polyhedral, or incidence geometry, and I will try to make the talk accessible to people with a broad range of backgrounds.
The main themes of the talk are (1) connecting the geometric properties of hypergraphs to their minimal matroids; (2) reducing the geometric invariants of these matroids to grid matroids; and (3) understanding the realizability of these matroids. 

Finally, I will mention the application to conditional independence models in statistics and will present several combinatorial questions and computational challenges around this problem.  This is based on joint works with Kevin Grace, Oliver Clarke, and Harshit J Motwani.

Given a fixed poset P, we say that a family F of subsets of [n] is P-free if it does not contain an (induced) copy of P. And we say that F is P-saturated if it is maximal P-free. How small can a P-saturated family be? The smallest such size is the induced saturation number of P, sat*(n,P). Even for very small posets, the question of the growth speed of sat*(n,P) seems to be hard. We present background on this problem and some recent results. 

Magnitude is an isometric invariant of metric spaces that has been introduced by Tom Leinster in 2010, and is currently the object of intense research, since it has been shown to encode many invariants of a metric space such as volume, dimension, and capacity. In 2018 I proposed a way to relate magnitude to persistent homology, one of the main techniques used in Topological Data Analysis. Building on this connection, in recent work, Hepworth and Govc introduced persistent magnitude, a numerical invariant of a filtered simplicial complex associated to a metric space. 

In this talk I will introduce magnitude, persistent magnitude, and further introduce alpha magnitude, a new invariant inspired by Hepworth and Govc’s notion of persistent magnitude, and establish some of its key properties. I will further explain how these invariants can be used to study datasets exhibiting self-similar properties, and compute some examples involving fractals. I will introduce all the concepts, and no previous knowledge on magnitude or persistent homology will be required

This talk is based on joint work with Sara Kalisnik and Miguel O’Malley, as well as the paper https://arxiv.org/abs/1807.01540

We consider the problem of managing the buffer of a shared-memory switch that transmits packets of unit value. A shared-memory switch consists of an input port, a number of output ports, and a buffer with a specific capacity. In each time step, an arbitrary number of packets arrive at the input port, each packet designated for one output port. Each packet is added to the queue of the respective output port. If the total number of packets exceeds the capacity of the buffer, some packets have to be irrevocably rejected. At the end of each time step, each output port transmits a packet in its queue and the goal is to maximize the number of transmitted packets.
The Longest Queue Drop (LQD) online algorithm accepts any arriving packet to the buffer. However, if this results in the buffer exceeding its memory capacity, then LQD drops a packet from whichever queue is currently the longest. The LQD algorithm was first introduced in 1991, and has shown to be 2-competitive in 2001. Although LQD remains the best known online algorithm for the problem, determining its true competitiveness is a long-standing open problem. We show that LQD is 1.707-competitive, establishing the first (2 − eps) upper bound for the competitive ratio of LQD, for a constant eps > 0, but there are still a number of outstanding questions.
(based on joint work with Antonios Antoniadis, Nicolaos Matsakis, and Pavel Vesely)

Let X be a family of subsets of a finite set E.  A matroid M on E is said to be an X-matroid if each set in X is a circuit in M.  We consider the problem of determining when there exists a unique maximal X-matroid in the weak order poset of all X-matroids on E (defined by putting M_1 ≤ M_2 if every independent set in M_1 is independent in M_2).

We present a new framework for designing worst-case to average-case reductions. For a large class of problems, it provides an explicit transformation of algorithms running in time T that are only correct on a small (subconstant) fraction of their inputs into algorithms running in time O(T log T) that are correct on all inputs.

Using our framework, we obtain such efficient worst-case to average-case reductions for fundamental problems in a variety of computational models; namely, algorithms for matrix multiplication, streaming algorithms for the online matrix-vector multiplication problem, and static data structures for all linear problems as well as for the multivariate polynomial evaluation problem.

Our techniques crucially rely on additive combinatorics. In particular, we show a local correction lemma that relies on a new probabilistic version of the quasi-polynomial Bogolyubov-Ruzsa lemma.

Joint work with Vahid Asadi, Alexander Golovnev, and Igor Shinkar.

There have been many attempts to extend the Tutte polynomial from graphs (or matroids) to embedded graphs (or delta-matroids). One which keeps many of the desirable properties of the Tutte polynomial, at least for graphs embedded in an orientable surface (or even delta-matroids), is sometimes known as the ribbon-graph polynomial and is essentially a two-variable polynomial specialization of Bollobas and Riordan's embedded graph polynomial.

We will review some of the many nice properties of the Tutte polynomial of a graph, namely that it is irreducible if and only if the graph (or matroid) is 2-connected and that it detects whether or not a graph (or matroid) is series-parallel and see that these properties extend to the ribbon graph polynomial.

No knowledge of either matroids or embedded graphs will be assumed!

Take k nodes uniformly at random in a n-node graph, and let k competing epidemics spread along edges at constant speed from these initial nodes, infecting nodes when they reach them. A node can only get infected by one of the epidemics, the first one that reaches it. We are interested in the sizes of the final territories of the k epidemics, which we call the Voronoi cells of the k initial nodes. In this joint work with Markus Heydenreich (Munich) and Alex Drewitz (Cologne), we prove limiting theorems for the sizes of the Voronoi cells of k nodes taken uniformly at random in an n-node random split tree.

A well-known phenomenon in random graphs is the phase-transition for connectivity, proved first by Erdős Rényi in 1959.

In this talk we will discuss a high-dimensional analogue of this phenomenon which we refer to as "homological connectivity".

Loosely speaking, homological connectivity is the point where the homology of a simplicial filtration stops changing.

The model we study is the Čech complex generated over a spatial Poisson point process.

We will show that there is a sequence of sharp phase transitions (for different degrees of homology) and also explore the behavior of the complex inside each critical window.

For any graph G, a family S of subsets of the edges of G is said to be a separating path system for G if both of the following hold. Firstly, for every pair of edges e,f in E(G), there is some P in S with the property that P contains e but not f. Secondly, every element of S is a path in G. A natural question is ‘how small can such a separating path system be?’.

This is joint work with David Munhá Correia and Benny Sudakov

Given a graph H, we say that a graph G is H-saturated if G is maximally H-free, meaning G contains no copy of H but adding any new edge to G creates a copy of H. The general saturation problem is to determine sat(n,H), the minimum number of edges in an H-saturated graph G on n vertices.
The special case when H is a triangle is straightforward – it is an easy exercise to show that sat(n,K_3) = n − 1 for n ≥ 1 and that the unique extremal graph is a star. Note that a star has many vertices of degree 1. One might ask what happens if we forbid such small degree vertices. We then have the more difficult problem of determining sat(n,K_3,t), the minimum number of edges in a triangle saturated graph G on n vertices that additionally has minimum degree at least t.
Day showed that for fixed t, sat(n,K_3,t) = tn − c(t) for large enough n, where c(t) is a constant depending on t. He proved the bounds 2^t t^(3/2) << c(t)  ≤  t^(t^(2t^2))​. We show that the order of magnitude of c(t) is 4^t / √t .
The order of magnitude of c(t) turns out to be intimately related to Bollobás’ celebrated Two Families Theorem. We end by presenting a conjectured generalisation of the Two Families Theorem, which, if proven, would allow one to extend these results from K_3 to general K_r.

Given an integer b>0. two players, called Maker and Breaker, play a game on a graph G. In each round, Breaker claims b unclaimed edges and then Maker claims one.  Maker wins if she can form a giant component (one with at least tn vertices, for some given constant t>0) from the edges she claimed. If the edges are all used up before she manages this, Breaker wins. Clearly there is either a winning strategy for Maker, or one for Breaker. The larger b is, the easier it is for Breaker to win.

Suppose that G is drawn randomly from the common random graph model G(n,p). Maker cannot possibly win unless $G$ has a giant component. The threshold value of p for which this giant component occurs (with high probability, i.e. close to 1) is 1/n.  So let p=c/n for c>1.  We can determine the smallest b for which Breaker wins with high probability.

A key part of our proof involves properties of the k-core of the random graph. I will give some background on this, as well as on Maker-Breaker games, and describe how the two have come together in this problem. This is joint work with Rani Hod, Michael Krivelevich, Tobias Muller and Alon Naor.

Based on joint works with Jacob Focke, Johannes Schmitt, and Philip Wellnitz.

An r-uniform tight cycle of length k>r is a hypergraph with vertices v_1,…,v_k and edges {v_i,v_(i+1),…,v_(i+r−1)} (for all i), with the indices taken modulo k. Sós, and independently Verstraëte, asked the following question: how many edges can there be in an n-vertex r-uniform hypergraph if it contains no tight cycles of any length? In this talk I will review some known results, and present recent progress on this problem.

Classical theorems of Dirac and of Ghouila-Houri establish the minimum degree theshold that guarantees the existence of a Hamilton cycle in a graph and in a directed graph respectively. (For oriented graphs, the degree threshold was established much more recently.) By imposing suitable regularity and/or connectivity conditions on graphs, one can lower the degree threshold at which a Hamilton cycle is guaranteed to appear, and this is quite well understood in the case of graphs. Conjectures of Kuhn and Osthus and of Jackson describe the corresponding situation for directed graphs and oriented graphs respectively. I will discuss approximate solutions to these conjectures.

This is based on joint work with Allan Lo and Mehmet Akif Yildiz.

A rainbow matching in an edge-coloured graph is a matching whose edges all have different colours. Let G be a group of order n and consider an edge-coloured complete bipartite graph, whose parts are each a copy of the group G, and the edge (x, y) gets coloured by the group element xy. We call this graph the multiplication table of G. For which groups G does the multiplication table of G have a rainbow matching? This is an old question in combinatorial group theory due to Hall and Paige, with close connections to the study of Latin squares. The problem has been resolved in 2009 with a proof relying on the classification of finite simple groups. In 2021, a "simpler" proof for large groups appeared, this time using tools from analytic number theory. We present a third resolution of this problem, again only for large groups, and using techniques from probabilistic combinatorics. The main advantage of our approach is that we are able to find rainbow matchings in random-like subgraphs of the multiplication table of G. This flexibility allows us to settle numerous longstanding conjectures in combinatorial group theory.

Exploring unknown environments is a fundamental task in many domains, e.g., robot navigation, network security, and internet search. In this talk, we give an overview of the current state of the art in online graph exploration in the fixed graph scenario. We provide details and lower bound examples of some algorithms such as the Nearest Neighbour (NN) algorithm with its competitive ratio of O(log n).
Afterwards, we design a general framework to robustify algorithms that carefully interpolates between a given algorithm and NN. We prove new performance bounds that leverage the individual good performance on particular inputs while establishing robustness to arbitrary inputs. This robustification schemes also allows us to initiate the study of a learning-augmented variant of the problem by adding access to machine-learned predictions and naturally integrating these predictions into the NN algorithm. This method significantly outperforms any known online algorithm if the prediction is of high accuracy while maintaining good guarantees when the prediction is of poor quality.

If a graph has large chromatic number, must it contain induced subgraphs with some particular local structure witnessing this property? A positive answer here would be useful in studying chi-boundedness, where an important strategy involves dropping down to an induced subgraph which still has high chromatic number but is structurally simpler (for instance, triangle-free). However, earlier this year, a string of papers by Carbonero, Hompe, Moore and Spirkl, and then Briański, Davies and Walczak provided fascinating constructions which in particular show that there are K_{p+1}-free graphs of large chromatic number in which every induced subgraph of large chromatic number contains a copy of K_p. In this talk, we'll discuss generalisations of this result replacing cliques by arbitrary graphs F with at least one edge.

Every finite graph has an associated sandpile group, which can be described in terms of chip-firing. A sandpile torsor algorithm is a map which associates each plane graph (i.e. planar embedding) with a free transitive action of its sandpile group on its spanning trees. We define a notion of consistency, which requires a torsor algorithm to be preserved with respect to a certain class of contractions and deletions. We then show that the rotor-routing sandpile torsor algorithm (which is equivalent to the Bernardi algorithm) is consistent. Furthermore, we demonstrate that there are only three other consistent algorithms, which all have the same structure as rotor-routing. This proves a conjecture of Klivans. Joint work with Ankan Ganguly.

When can (the edge-set of) a graph G be decomposed into copies of a given graph H? This question goes all the way back to Euler; despite this, the setting where the number of vertices in G and H are comparable is not yet well-understood. I will talk about the resolution of a conjecture of Ringel from 1963 where G is the complete graph on 2n+1 vertices and H is any given tree with n edges. This is joint work with Peter Keevash; the conjecture was independently resolved by Montgomery, Pokrovskiy and Sudakov.

The problem of self-avoiding walks in a domain (e.g. a finite-size square or rectangular subset of the square lattice, or a triangular or hexagonal subset of the hexagonal lattice) is both a combinatorial problem of considerable interest, and has applications to telecommunications. Designing efficient algorithms to study this problem by exact enumeration has recently led to spectacular advances in reducing computational complexity. I will report on recent joint work with Iwan Jensen (Flinders University) and Aleks Owczarek (Melbourne) which has enabled us to conjecturally determine the asymptotics very precisely. Furthermore, in current work we have proved the equivalence of this problem to the seemingly unrelated combinatorial problem of gerrymander sequences, and have consequently been able to give very precise asymptotic estimates for that problem too.

Many classical theorems of additive combinatorics state that a certain sumset is large.  For example, the Cauchy-Davenport theorem gives a lower bound on the size of
the sumset A+B for two subsets A and B of the cyclic group of order p.  What would happen if we insisted that, in forming the sums, we were only allowed to use a bounded number of elements of B?  How close could we get to the lower bound of |A|+|B|-1 given by the Cauchy-Davenport theorem, for example?  (Joint work with Bela Bollobas and Marius Tiba.)

Erdős and McKay conjectured that if all homogeneous sets in an n-vertex graph are of order O(log n) then the graph contains induced subgraphs of each size from {0,1,…,Ω(n^2)}. In this talk I will outline a proof of a bipartite version of this conjecture: if all balanced homogeneous sets in an n×n bipartite graph are of order O(log n) then the graph contains induced subgraphs of each size from {0,1,…,Ω(n^2)}.

Based on joint work with Laurentiu Ploscaru.

For a fixed poset P, a family F of subsets  of [n] is induced P-saturated if F does not contain an induced copy of P, but for every subset S of [n] such that S \notin F, then P is an induced subposet of F \cup {S}.  The size of the smallest such family F is denoted by sat*(n,P). 

In recent years, there has been a growing interest in tackling the problem of determining sat*(n,P).  Keszegh,  Lemons,  Martin, Pálvölgyi and  Patkós [Journal of Combinatorial Theory Series A, 2021] proved that there is a dichotomy of behaviour for this parameter: given any poset P, either sat*(n,P)=O(1) or sat*(n,P) ≥ log _2 n.  Furthermore, they conjectured that either sat*(n,P)=O(1)  or sat*(n,P) ≥ n+1.

In this talk we present some progress on this conjecture, namely that either sat*(n,P)=O(1) or sat*(n,P) ≥ 2 \sqrt(n-2).  This is joint work with Simón Piga, Maryam Sharifzadeh and Andrew Treglown. 

I will describe the game of cops-and-robbers and then its generalisation to multilayer graphs. In this setting, a graph consists of a single set of vertices with multiple (potentially intersecting) edge sets.  We allow the cops and robber to move only on their assigned layer, and ask of the cops can be guaranteed to catch the robber in finite time.   Using several examples, I’ll show that initial intuition about the best way to allocate cops to layers is not always correct.  I will outline arguments showing that the number of cops required to catch a robber in a multilayer graph is neither bounded from above nor below by any function of the cop numbers of the individual layers.  Additionally, we’ll talk about a question of worst-case division of a simple graph into layers: given a simple graph G, what is the maximum number of cops required to catch a robber over all multilayer graphs where each edge of G is in at least one layer and all layers are connected? For cliques, suitably dense random graphs, and graphs of bounded treewidth, we have determined this parameter up to multiplicative constants.
Lastly I’ll outline a multilayer variant of Meyniel's conjecture, and show the existence of an infinite family of graphs whose multi-layer cop number is bounded from below by a constant times n / log n, where n is the number of vertices in the graph.

We consider directed graphs without loops, and rules that select a subset of the vertices in any graph. A selection rule is impartial if the selection of a vertex is independent of its outgoing edges and we also aim to make our selection rule competitive, that is, select vertices with high indegree. This models peer selection where vertices are candidates and directed edges represent votes. We wish to select highly voted candidates such that a candidate cannot influence their own selection.

Given a properly coloured complete graph, a natural problem to consider is that of finding large and spanning rainbow substructures. Looking at matchings, we know that in some optimal proper colourings of the complete balanced bipartite graph one can't obtain a perfect rainbow matching. However, proving the Ryser-Brualdi-Stein Conjecture for large even n, Montgomery recently showed that one can always guarantee a rainbow matching missing at most two vertices in any optimal colouring. Looking at paths in properly coloured (non-bipartite) complete graphs, again it is known that we can't guarantee a rainbow Hamilton path. However, Andersen conjectured that one can always find a rainbow path avoiding at most one vertex. In this talk we'll discuss these problems in more detail, including what is already known and some current work in progress with Alistair Benford and Richard Montgomery.

We search for the class of digraphs whose obstruction sets, in the digraph homomorphism problem, consist of caterpillars. The conjecture being that this class of digraphs should be the same as the class of digraphs that are preserved by majority and set function polymorphisms. This is  work in progress.

The simplest Cops and Robbers game is as follows. Both the cop and robber are placed on a graph and then they alternately take turns where they either move to a neighbouring vertex or remain at the same vertex. The cop wins if, at some point, he is at the same vertex as the robber – the robber wins if he can ensure this never happens.

If the graph is finite then it is easy to show that a graph is cop-win if and only if it is constructible, meaning that the graph can be obtained from the one-point graph by adding dominated vertices one at a time.

But if the graph is infinite then the situation is more complicated. We will discuss what happens, including the first example of a cop-win graph that is not constructible.

Joint work with Maria Ivan and Imre Leader

Deciding how to allocate the seats of a deliberative assembly is one of the most fundamental problems in the political organization of societies, and has been widely studied over already two centuries. The idea of proportionality is at the core of most approaches to tackle this problem, and this notion is captured by the divisor methods, such as the Jefferson/D'Hondt method. In a seminal work, Balinski and Demange extended the single-dimensional idea of divisor methods to the setting in which the seat allocation is simultaneously determined by two dimensions, and proposed the so-called biproportional apportionment method. The method, currently used in several electoral systems, is however limited to two dimensions and the question of extending it is considered to be an important problem both theoretically and in practice.

In this work we initiate the study of multidimensional proportional apportionment. We first formalize a notion of multidimensional proportionality that naturally extends that of Balinski and Demange. By means of analyzing an appropriate integer linear program we are able to prove that, in contrast to the two-dimensional case, the existence of multidimensional proportional apportionments is not guaranteed and deciding their existence is NP-complete. Interestingly, our main result asserts that it is possible to find approximate multidimensional proportional apportionments that deviate from the marginals by a small amount. The proof arises through the lens of discrepancy theory, mainly inspired by the celebrated Beck-Fiala Theorem. We finally evaluate our approach by using the data from the recent 2021 Chilean Constitutional Convention election.

This is joint work with José Correa and Victor Verdugo.

Suppose that the vertices of a regular graph are coloured red and blue with an equal number of each (we call this a balanced colouring). Since the graph is undirected, the number of edges from a red vertex to a blue vertex is clearly the same as the number of edges from a blue vertex to a red vertex. However, if instead we count walks of length 2, then this symmetry disappears. How extreme can this asymmetry be?

More precisely, given a d-regular graph, for which pairs (x,y) is there a balanced colouring for which the probability that a random walk starting from a red vertex stays within the red class for at least 2 steps is x, and the corresponding probability for blue is y?

We will discuss a variety of questions and results on this including a general bound for d-regular graphs, an exact (asymptotic) answer for the 2-dimensional torus, and many open problems.

Joint work with Ron Gray.

How much local information is needed to reconstruct a graph? In the graph shotgun assembly problem, we are given the balls of radius r around the vertices of a graph and we are asked to identify the graph. We will consider this problem for the Erdős-Rényi random graph G(n,p) for a wide range of values of r and discuss the thresholds on p for when the graph is or is not reconstructible with high probability. In particular, we will give the threshold for each r ≥ 3.

A typical graph contains cliques and independent sets of no more than logarithmic size.  The Erdos-Hajnal Conjecture asserts that if we forbid some induced subgraph H then we can do much better: the conjecture claims that there is some c=c(H)>0 such that every H-free graph G contains a clique or independent set of size at least |G|^c.  The conjecture looks far out of reach, and is only known for a small family of graphs.  We will survey some recent progress.

Joint work with Tung Nguyen and Paul Seymour.

The Helly number of a set in the plane is the smallest N such that the following is true. If any N members of a finite family of convex sets contains a point of S, then there is a point of S which is contained in all members of the family. An exponential lattice with base x consists of points whose coordinates are positive integer powers of x. We prove lower and upper bounds on Helly numbers of exponential lattices in terms of x, and we determine their values exactly in some cases. We also consider asymmetric exponential lattices, and characterise those that have finite Helly numbers. Joint work with Gergely Ambrus, Martin Balko, Attila Jung and Márton Naszódi.

In this talk, we will discuss the evolution of classes on non-monotone dynamics on a binomial random graph. In particular, we will start with majority dynamics either with two or more states. There, each vertex updates their state adopting the state that has the majority among its neighbours.

We will continue with generalisations of this dynamics in the context of population games. Assume that a 2-player symmetric game with 2 strategies is played between the adjacent members of the vertex set.

(In this context the states are the strategies.) Players/vertices update their strategies synchronously: at each round, each player selects the strategy that is the best response to the current set of strategies its neighbours play. We show that such a system reduces to generalised majority and minority dynamics. We show rapid convergence to unanimity for p in a range that depends on a certain characteristic of the payoff matrix. In the presence of a certain type of bias in the payoff matrix, we determine a sharp threshold on p above which the largest connected component reaches unanimity with high probability, and below which this does not happen.

This is joint work with Calina Durbac and Jordan Chellig

Matroid intersection is a classical optimization problem where, given two matroids over the same ground set, the goal is to find the largest common independent set. We show how to construct a certain "sparsifer": a subset of elements, of size O(|S^{opt}| \cdot 1/epsilon), where S^{opt} denotes the optimal solution, that is guaranteed to contain a 3/2 + epsilon approximation, while guaranteeing certain robustness properties. We call such a small subset a \emph{Density Constrained Subset} (DCS), which is inspired by the Edge-Degree Constrained Subgraph (EDCS) [Bernstein and Stein, 2015], originally designed for the maximum cardinality matching problem in a graph. Our proof is constructive and hinges on a greedy decomposition of matroids, which we call the \emph{density-based decomposition}. We show that this sparsifier has certain robustness properties that can be used in one-way communication and random-order streaming models.

A bisection in a graph is a cut in which the number of vertices in the two parts differ by at most 1. In this paper, we give lower bounds for the maximum weight of bisections of edge-weighted graphs with bounded maximum degree. We show a tight lower bound for the maximum size of bisections in 3-regular graphs. We also consider edge-weighted triangle-free subcubic graphs and show that a much better lower bound (than for edge-weighted subcubic graphs) holds for such graphs if we exclude K1,3. This is joint work with Gregory Gutin, Anders Yeo, and Yacong Zhou.

Graph width parameters play an important role in the complexity of algorithms on graphs. For example, the metatheorems of Courcelle and coauthors typically say something like the following: an evaluation problem can be solved in polynomial time on some graph class if (1) the problem can be expressed in some restricted form of logic, and (2) every graph in the class has a suitable bounded width parameter.

One well-known parameter is tree-width: Robertson and Seymour showed that there is a single fundamental obstruction to minor-closed classes having bounded tree-width: the class of planar graphs. (That is, a class has bounded tree-width if and only if it forbids some planar graph as a minor). More recently, Geelen, Kwon, McCarty and Wollan showed that there is a similar single fundamental obstruction to vertex-minor-closed classes having bounded rank-width: the class of circle graphs.

For clique-width (which is strongly related to rank-width) and hereditary graph classes (closed under taking induced subgraphs), the situation is much worse.  Two minimal hereditary classes with unbounded clique-width were found in 2011 by Lozin, and in the 13 years since more have been uncovered, including a countably infinite collection in 2018. In this talk, I will present a framework that can be used to construct most of the known minimal classes, and also uncountably many more. Time permitting, I will also discuss how the hunt for minimal classes is not the only task if one wanted to establish a complete characterisation.

This talk is based on joint work with Dan Cocks.

A linear graph code is a family C of graphs on n vertices such that the symmetric difference of the edge sets of any two graphs in C is also the edge set of a graph in C. In the talk, we will investigate the maximal size of a graph code that does not contain a copy of a fixed graph H. There are graphs H that are not contained in linear codes of size 2^{ \binom{n}{2} } / exp(sqrt(log n)), but we will show that for almost all graphs H with an even number of edges, there exists eps_H>0 such that a linear graph code without copy of H can contain at most 2^{\binom{n}{2}} / n^{eps_H} graphs.

In 1990 Jaeger, Vertigan and Welsh published a paper "On the computational complexity of the Jones and Tutte polynomials" in Math. Proc. Cambridge Philos. Soc.  In doing so they started a flourishing line of enquiry that brought together researchers in combinatorics, theoretical computer science, probability and statistical physics.  The 157 citations listed on MathSciNet attest to the impact of the paper.  Dominic Welsh died on 30th November, and this talk is a small contribution to his memory.

The Tutte polynomial is a two-variable polynomial associated with a matroid.  If you are not a fan of matroids, don't worry:  the only ones that will arise in this talk are very concrete, specifically, graphic matroids (associated with undirected graphs) and binary matroids (associated with sets of vectors over GF(2)).  I'll define the Tutte polynomial, and provide interpretations of the polynomial evaluated at certain points and along certain hyperbolas in R^2.  I'll survey the results of Jaeger, Vertigan and Welsh, which give a complete picture of the computational complexity of exactly evaluating the Tutte polynomial in the graphic case.  I'll then describe a couple of substantial fairly recent advances:  (i) an efficient algorithm by Anari, Liu, Oveis Gharan and Vinzant for approximating the Tutte polynomial in a region of R^2 that includes network reliability;  (ii) a proof by Noble of a "lost theorem" of Vertigan concerning the number of bases of a binary matroid.   

Understanding mechanisms for generating random graph models with properties of real networks is an important topic in complex networks. Although there is a large literature on random graph models with a power-law degree distribution, a positive clustering coefficient or with a giant component, much work remains to be done to generate random graph models with several of these properties. In this talk we will discuss some results about random graph models with power-law degree distribution and giant component. At the end, we will present some work in progress about random graph models with power-law degree distribution and positive clustering coefficient.

Many cities around the world allocate a part of their budget based on residents’ votes, following a process known as participatory budgeting. It is important to understand which outcomes of this process should be viewed as fair, and whether fair outcomes could be computed efficiently. We summarise recent progress on this topic. We first focus on a special case of participatory budgeting where all candidate projects have the same cost (known as multiwinner voting), formulate progressively more demanding notions of fairness for this setting, and identify efficiently computable voting rules that satisfy them. We then discuss the challenges of extending these ideas to the general model.

We prove for large graphs the following conjecture of Alavi, Boals Chartrand, Erdős and Oellermann from 1987: any graph with (m+1)m/2 edges can be decomposed into graphs G_1,...,G_m such that G_i has exactly i edges and is isomorphic to a subgraph of G_{i+1}. This is joint work with Shoham Letzter, Alexey Pokrovskiy and Benny Sudakov.

We give a strongly polynomial algorithm for minimum cost generalized flow, and hence for optimizing any linear program with at most two non-zero entries per row, or at most two non-zero entries per column. Our result can be viewed as progress towards understanding whether all linear programs can be solved in strongly polynomial time, also referred to as Smale's 9th problem.

Our approach is based on the recent primal-dual interior point method (IPM) due to Allamigeon, Dadush, Loho, Natura and Végh (FOCS '22). They show that the number of iterations needed by the IPM can be bounded in terms of the straight line complexity of the central path; roughly speaking, this is the minimum number of pieces of any piecewise linear curve that multiplicatively approximates the central path. As our main contribution, we show that the straight line complexity of any minimum cost generalized flow instance is polynomial in the number of arcs and vertices. By a reduction of Hochbaum, the same bound applies to any linear program with at most two non-zeros per column or per row.  Further, we demonstrate how to handle initialization, and how to ensure that the bit complexity of each iterate remains bounded during the execution of the algorithm.

Joint work with Daniel Dadush, Zhuan Khye Koh, Bento Natura and László Végh.

Given a family F of graphs, we say that a graph G is F-saturated if it is maximally F-free, meaning G does not contain a graph in F but adding any new edge to G creates a graph in F. We then define sat(n,F)$  to be the minimum number of edges in an F-saturated graph on n vertices. In 1986, Kászonyi and Tuza showed that sat(n,F)=O(n)$ for all families F and Tuza conjectured that for singleton families sat(n,F)/n converges. Tuza's Conjecture remains wide open. In this talk, I will discuss recent results about the asymptotic behaviour of sat(n,F)$, mostly in the sparse regime sat(n,F) ≤ n+o(n)$, in each of the cases when F is a singleton, when F is finite and when F is possibly infinite.

Joint work with Andrea Freschi.

In stochastic function evaluation, one is given a (succinctly represented) Boolean function and n (typically independent) random variables with known probability distributions. Each of the variables can be queried at known cost, which reveals the value of the respective variable. The goal is to efficiently compute a strategy for querying variables that determines the value of the function applied to these variables at (approximately) minimum expected cost.
In the first part of this talk, we consider score-classification functions (Gkenosis, Grammel, Hellerstein, and Kletenik; 2018). Here, variables are Boolean and the function value only depends on which given subinterval of [0, n] the total number of successes is in. We give simple non-adaptive constant-factor approximation algorithms. This part is joint work with Benedikt Plank.
In the second part of this talk, we consider a setting where the random variables correspond to votes, their values correspond to candidates, and the value of the function is the outcome of an election. We give adaptive constant-factor approximations for the absolute-majority and relative-majority versions of this problem. This part is joint work with Lisa Hellerstein and Naifeng Liu.

The Turán density t(s,r) is the asymptotically smallest edge density of a r-graph in which every set of s vertices contains at least one edge. The question of estimating this function received a lot of attention after it was first raised by Turán in 1941. A trivial lower bound is t(s,r)\ge 1/{s\choose s−r). In the early 1990s, de Caen conjectured that t(r+1,r) grows faster than O(1/r) and offered 500 Canadian dollars for resolving this question.

I will present a construction disproving this this conjecture by showing more strongly that for every integer R there is C such that t(r+R,r)\le C/{r+R\choose R}, that is, the trivial lower bound is tight for every fixed R up to a multiplicative constant C=C(R).

A classic result due to Vizing states that every graph with maximum degree ∆ has a proper edge- colouring using at most ∆ + 1 colours. In this talk, we will discuss a natural generalisation of this problem in which the task is to properly edge-colour several graphs simultaneously. More precisely, given graphs G1, . . . , Gk on the same vertex set, we are interested in the minimum number of colours needed to colour the edges of G1 ∪ · · · ∪ Gk so that the colouring induced on each individual graph Gi is proper; we denote this number by χ′(G1, . . . , Gk). This problem was proposed by Cabello and was studied by Bousquet and Durain and subsequently by Cambie. We improve upon their work and show that, for any graphs G1 and G2 with maximum degree at most ∆, we have χ′ (G1 , G2 ) = (1 + o(1))∆ as ∆ → ∞. Our ideas generalise to an arbitrary number of graphs, again yielding asymptotically tight results. In the process, we uncover some interesting connections between the simultaneous colouring problem and some old problems in combinatorics.

Suppose we fix a finite collection S of points in R^d, which we regard as the locations of voters. Two players (candidates), Alice and Bob, are playing the following game. Alice goes first and chooses any point A in R^d. Bob can then choose any point B in R^d, knowing what A is. A voter V positioned at x will vote for Alice (resp. Bob) if d(x,A)<d(x,B) (resp. d(x,B)<d(x,A)), where d denotes Euclidean distance. If x is equidistant from A and B, V will not vote for anyone. The candidate with the greatest number of votes wins. If they both get the same number of votes, then (by convention) Alice is declared the winner.

When d=1, it is easy to see that Alice can always win this game by choosing A to be the median of S. This observation was first made by Hotelling in the 1920s; he referred to this model (in one dimension) as the Median Voter Model. When d > 1 however, the game is sometimes an Alice win and sometimes a Bob win, depending on the structure of S. We completely characterize the sets S for which Alice wins in dimensions > 1, showing that the game is usually won by Bob, unless S has a specific structure.

This project originated in a question that I posed at a open problem session in 2020: are there uncountably many minor-closed matroid classes that are well-quasi-ordered under the minor relation? Rutger Campbell quickly showed that the answer is yes. With further discussion, we realised that this exemplifies a much more general problem.

A partially ordered set (poset) is well-founded if it does not contain an infinite descending chains. A down-set in a poset is a set of elements that is closed downwards: meaning that if x is in the set and x' < x, then x' too is in the set. If a poset is well-founded, then a down-set is well-quasi-ordered if it does not contain any infinite antichains. Given a countable well-founded poset, we ask when there are uncountably many well-quasi-ordered down-sets. We now have necessary and sufficient conditions which answer this question and which apply under quite mild conditions (in particular, they apply to the matroid minor-order, and to many other orders which arise naturally in graph theory and combinatorics).

This is joint work with Rutger Campbell (Institute of Basic Science, Daejeon, South Korea)

The `semirandom graph process', introduced in 2020, has attracted a good deal of study. I will discuss the 2-offer 3-uniform semirandom hypergraph model on n vertices. Here, at each step, we are presented with 2 uniformly random vertices. We choose any other vertex, thus creating a hyperedge of size 3. The aim is to create a desired structure, in our case a perfect matching or loose Hamilton cycle, as quickly as possible. We show strategies which in Theta(n) steps succeed asymptotically almost surely; up to constants this is clearly best possible. The challenges with hypergraphs, and our methods, are qualitatively different from what has been seen for semirandom graphs. Much of our analysis is done on an auxiliary graph that is a uniform k-out subgraph of a random bipartite graph, and this tool may be useful in other contexts.

This is work with Mike Molloy and Pawel Pralat.

Given a k-uniform hypergraph H on n vertices with an r-colouring of its edges, we look for a minimum l-degree condition that guarantees the existence of a perfect matching in H that has more than n/rk edges in one colour. We call this a colour-bias perfect matching.

For 2-coloured graphs, a result of Balogh, Csaba, Jing and Pluhár yields the minimum degree threshold that ensures a perfect matching of significant colour-bias. In this talk, I will present an analogous of this result for k-uniform hypergraphs. More precisely, for each 1<=l<k and r>=2 we determined the minimum l-degree threshold that forces a perfect matching of significant colour-bias in an r-edge-coloured k-uniform hypergraph.

This is joint work with J. Balogh, H. Hàn, R. Lang, J. P. Marciano, M. Pavez-Signé, N. Sanhueza-Matamala and A. Treglown.

We consider the following activation process: start from a tree and, at every step, add an edge if it creates a triangle. Clearly, any spanning tree activates the complete graph. Is it possible to activate the binomial random graph G(n,p) from some spanning tree? Using the fact that the possibility of activating a graph implies simple connectivity of the 2-dimensional clique complex of this graph, we show that the threshold for the above property equals n^{-1/3+o(1)}. We further develop a new approach to analyse this process inspired by Gromov’s results on word hyperbolicity of finitely generated groups. It allows us to find the threshold up to a constant factor for spanning trees of bounded depth.
Joint work with Asaf Cohen-Antonir, Yuval Peled, Asaf Shapira, and Mykhaylo Tyomkyn.

The Brunn-Minkowski inequality is a fundamental result in convex geometry and analysis, closely related to the isoperimetric inequality. It states that for (open) sets A and B in R^d, we have |A+B|^{1/d} ≥ |A|^{1/d}+|B|^{1/d}. Here A+B={x+y : x \in A, y \in B}. Equality holds if and only if A and B are homothetic and convex sets in R^d. 

The Prekopa-Leindler inequality is a functional generalization of the Brunn-Minkowski inequality with important applications to high dimensional probability theory. If t \in (0,1) and f,g,h : R^d -> R_+ are continuous functions with bounded support such that h(z) = \sup_{z = tx + (1-t)y} f^t(x) g^{1-t}(y),  then  \int h dx ≥ (\int f dx)^t  (\int g dx)^{1-t}. Equality holds if and only if f and g are homothetic (i.e. f = ag(x+b)) and log-concave (i.e. \log(f) is concave). The Borell-Brascamp-Lieb inequality is a strengthening of the Prekopa-Leindler inequality, replacing the geometric mean with other means. 

The stability of these inequalities has been intensely studied lately. The stability of the Brunn-Minkowski inequality states that if we are close to equality, then A and B must be close to being homothetic and convex. Similarly, the stability of the Prekopa-Leindler and Borell-Brascamp-Lieb inequalities states that if we are close to equality, then f and g must be close to being homothetic and concave. In this talk, we present sharp stability results for the Brunn-Minkowski, Prekopa-Leindler and Borell-Brascamp-Lieb inequalities, establishing the exact dependency between the two notions of closeness, thus concluding a long line of research on these problems. 

This talk is based on joint work with Alessio Figalli and Peter van Hintum.

In 1971, Roger Penrose published a brilliant conference proceedings a paper "Applications of negative-dimensional tensors". In this paper Penrose uses his "abstract tensor systems" to study edge 3-colourings of planar graphs (and hence the Four-Colour Theorem which was still a conjecture at that time). Penrose's work was brought into the combinatorial fold in 80's and 90's by researchers including P. Martin and F. Jaeger, but it was the work of M. Aigner in the early 2000's that consolidated Penrose's ideas with several excellent articles on the "Penrose polynomial" of a plane graph. A key property of the polynomial is that its evaluation at 3 counts the number of edge 3-colourings of the graph --- an interpretation of Penrose's original result.

In the early 2010s J. Ellis-Monaghan and I extended the Penrose polynomial to graphs embedded in higher genus surfaces. By doing this we were able to 'complete the picture' for some of Aigner's results, but the connection with edge 3-colourings was lost for higher genus surfaces. In the mid 2010's L. Kauffman introduced a different extension of the Penrose polynomial which is defined for graphs equipped with a perfect matching, but it does count edge 3-colourings for all graphs. A couple of years later, Kaufman and Ellis-Monaghan and I discovered a third extension of the Penrose polynomial that counts edge 3-colourings for graphs embedded in any surface. Then another Penrose polynomial appeared as a Poincare polynomial in S. Baldridge and B. McCarty's 2023 work on a TQFT approach to graph colouring. So just how many Penrose polynomials are there?

In this talk I'll offer an introduction to the Penrose polynomial of graph embedded in a surface, with a particular focus on its recent extensions to graphs embedded in higher genus surfaces and graphs with perfect matchings.

When selecting a subset of candidates (a so-called committee) based on the preferences of voters, proportional representation is often a major desideratum. When going beyond simplistic models such as party-list or district-based elections, it is surprisingly challenging to capture proportionality formally. As a consequence, the literature has produced numerous competing criteria of when a selected committee qualifies as proportional. Two of the most prominent notions are Dummett's proportionality for solid coalitions (PSC) and Aziz et al.'s extended justified representation (EJR). Both definitions guarantee proportional representation to groups of voters who have very similar preferences; such groups are referred to as solid coalitions by Dummett and as cohesive groups by Aziz et al. However, these notions lose their bite when groups are only almost solid or almost cohesive. In this paper, we propose proportionality axioms that are more robust than their existing counterparts, in the sense that they guarantee representation also to groups that do not qualify as solid or cohesive. Importantly, we show that these stronger proportionality requirements are always satisfiable. Another important advantage of our novel axioms is that their satisfaction can be easily verified: Given a committee, we can check in polynomial time whether it satisfies the axiom or not. This is in contrast to many established notions like EJR, for which the corresponding verification problem is known to be intractable.

In the setting with approval preferences, we propose a robust and verifiable variant of EJR and a simple greedy procedure to compute committees satisfying it. We show that our axiom is considerably more discriminating in randomly generated instances compared to EJR and other existing axioms. In the setting with ranked preferences, we propose a robust variant of Dummett's PSC. In contrast to earlier strengthenings of PSC, our axiom can be efficiently verified even for general weak preferences. In the special case of strict preferences, our notion is the first known satisfiable proportionality axiom that is violated by the Single Tranferable Vote (STV). In order to prove that our axiom can always be satisfied, we extend the notion of priceability to the ranked preferences setting. We also discuss implications of our results for participatory budgeting, querying procedures, and to the notion of proportionality degree.

Based on joint work with Jannik Peters.
https://arxiv.org/abs/2302.01989

In 1995,  Erdős and Gyárfás proved that for every red-blue edge-colouring of the complete graph on n vertices, there exists a collection of at most 2\sqrt{n} monochromatic paths, all of the same colour, which together cover the entire vertex set. They conjectured that 2\sqrt{n} can be replaced by \sqrt{n}. In joint work with Alexey Pokrovskiy and Leo Versteegen, we prove this to be true for all large n.