Past Combinatorial Theory Seminar

We study random (simple) graphs with given vertex degrees, in the sparse case where the average degree is bounded. Assume also that the second moment of the vertex degree is bounded. The standard method then is to use the configuration model to construct a random multigraph and condition it on
being simple.

This works well for results of the type that something holds with high probability, or that something converges in probability, but it does not immediately apply to convergence in distribution, for example asymptotic normality. (Although this has been done by special arguments in a couple of cases, by Janson and Luczak and by Riordan.) A typical example is the recent result by Barbour and Röllin on asymptotic normality of the size of the giant component of the multigraph (in the supercritical case); it is an obvious conjecture that the same results hold for the random simple graph.

We discuss two new approaches to this, both based on old methods. Both apply to the size of the giant component, using rather minor special arguments.

One approach uses the method of moments to obtain joint convergence of the variable of interest together with the numbers of loops and multiple edges
in the  multigraph.

The other approach uses switchings to modify the multigraph and construct a simple graph. This simple random graph will not have a uniform distribution,
but almost, and this is good enough.

A well-known conjecture in computer science and statistical physics is that Glauber dynamics on the set of k-colorings of a graph G on n vertices with maximum degree \Delta is rapidly mixing for k \ge \Delta+2. In 1999, Vigoda showed rapid mixing of flip dynamics with certain flip parameters on the set of proper k-colorings for k > (11/6)\Delta, implying rapid mixing for Glauber dynamics. In this paper, we obtain the first improvement beyond the (11/6)\Delta barrier for general graphs by showing rapid mixing for k > (11/6 - \eta)\Delta for some positive constant \eta. The key to our proof is combining path coupling with a new kind of metric that incorporates a count of the extremal configurations of the chain. Additionally, our results extend to list coloring, a widely studied generalization of coloring. Combined, these results answer two open questions from Frieze and Vigoda’s 2007 survey paper on Glauber dynamics for colorings. 

This is joint work with Michelle Delcourt and Luke Postle.

There was major progress on perfect graphs in the early 2000's: Chudnovsky, Robertson, Thomas and I proved the ``strong perfect graph theorem'' that a graph is perfect if and only if it has no odd hole or odd antihole; and Chudnovsky, Cornuejols, Liu, Vuscovic and I found a polynomial-time algorithm to test whether a graph has an odd hole or odd antihole, and thereby test if it is perfect. (A ``hole'' is an induced cycle of length at least four, and an ``antihole'' is a hole in the complement graph.)

What we couldn't do then was test whether a graph has an odd hole, and this has stayed open for the last fifteen years, despite some intensive effort. I am happy to report that in fact it can be done in poly-time (in time O(|G|^{12}) at the last count), and in this talk we explain how.

Joint work with Maria Chudnovsky, Alex Scott, and Sophie Spirkl.

The Erdos-Hajnal Theorem asserts that non-universal graphs, that is, graphs that do not contain an induced copy of some fixed graph H, have homogeneous sets of size significantly larger than one can generally expect to find in a graph. We obtain two results of this flavor in the setting of r-uniform hypergraphs.

1. A theorem of R\"odl asserts that if an n-vertex graph is non-universal then it contains an almost homogeneous set (i.e one with edge density either very close to 0 or 1) of size \Omega(n). We prove that if a 3-uniform hypergraph is non-universal then it contains an almost homogeneous set of size \Omega(log n). An example of R\"odl from 1986 shows that this bound is tight.

2. Let R_r(t) denote the size of the largest non-universal r-graph G so that neither G nor its complement contain a complete r-partite subgraph with parts of size t. We prove an Erd\H{o}s--Hajnal-type stepping-up lemma, showing how to transform a lower bound for R_r(t) into a lower bound for R_{r+1}(t). As an application of this lemma, we improve a bound of Conlon-Fox-Sudakov by showing that R_3(t) \geq t^{ct).

Joint work with M. Amir and A. Shapira

The extremal number ${\rm ex}(n,F)$ of a graph $F$ is the maximum number of edges in an $n$-vertex graph not containing $F$ as a subgraph. A real number $r \in [0,2]$ is realisable if there exists a graph $F$ with ${\rm ex}(n , F) = \Theta(n^r)$. Several decades ago, Erdős and Simonovits conjectured that every rational number in $[1,2]$ is realisable. Despite decades of effort, the only known realisable numbers are $0,1, \frac{7}{5}, 2$, and the numbers of the form $1+\frac{1}{m}$, $2-\frac{1}{m}$, $2-\frac{2}{m}$ for integers $m \geq 1$. In particular, it is not even known whether the set of all realisable numbers contains a single limit point other than two numbers $1$ and $2$.

We discuss some progress on the conjecture of Erdős and Simonovits. First, we show that $2 - \frac{a}{b}$ is realisable for any integers $a,b \geq 1$ with $b>a$ and $b \equiv \pm 1 ~({\rm mod}\:a)$. This includes all previously known ones, and gives infinitely many limit points $2-\frac{1}{m}$ in the set of all realisable numbers as a consequence. Secondly, we propose a conjecture on subdivisions of bipartite graphs. Apart from being interesting on its own, we show that, somewhat surprisingly, this subdivision conjecture in fact implies that every rational number between 1 and 2 is realisable.

This is joint work with Jaehoon Kim and Hong Liu.

We continue the study by Melo and Winter [arXiv:1712.01763, 2017] on the possible intersection sizes of a $k$-dimensional subspace with the vertices of the $n$-dimensional hypercube in Euclidean space. Melo and Winter conjectured that all intersection sizes larger than $2^{k-1}$ (the “large” sizes) are of the form $2^{k-1} + 2^i$. We show that this is almost true: the large intersection sizes are either of this form or of the form $35\cdot2^{k-6}$ . We also disprove a second conjecture of Melo and Winter by proving that a positive fraction of the “small” values is missing.

The classical theory of Erdős–Rényi random graphs is concerned primarily with random subgraphs of $K_n$ or $K_{n,n}$. Lately, there has been much interest in understanding random subgraphs of other graph families, such as regular graphs.

We study the following problem: Let $G$ be a $k$-regular bipartite graph with $2n$ vertices. Consider the random process where, beginning with $2n$ isolated vertices, $G$ is reconstructed by adding its edges one by one in a uniformly random order. An early result in the theory of random graphs states that if $G=K_{n,n}$, then with high probability a perfect matching appears at the same moment that the last isolated vertex disappears. We show that if $k = Ω(n)$, then this holds for any $k$-regular bipartite graph $G$. This improves on a result of Goel, Kapralov, and Khanna, who showed that with high probability a perfect matching appears after $O(n \log(n))$ edges have been added to the graph. On the other hand, if $k = o(n / (\log(n) \log (\log(n)))$, we construct a family of $k$-regular bipartite graphs in which isolated vertices disappear long before the appearance of perfect matchings.

Joint work with Roman Glebov and Zur Luria.
 

How long a monotone path can one always find in any edge-ordering of the complete graph $K_n$? This appealing question was first asked by Chvatal and Komlos in 1971, and has since attracted the attention of many researchers, inspiring a variety of related problems. The prevailing conjecture is that one can always find a monotone path of linear length, but until now the best known lower bound was $n^{2/3−o(1)}$, which was proved by Milans. This talk will be
about nearly closing this gap, proving that any edge-ordering of the complete graph contains a monotone path of length $n^{1−o(1)}$. This is joint work with Bucic, Kwan, Sudakov, Tran, and Wagner.

Let $G$ be a regular graph of degree $d$ and let $A\subset V(G)$. Say that $A$ is $\eta$-closed if the average degree of the subgraph induced by $A$ is at least $\eta d$. This says that if we choose a random vertex $x\in A$ and a random neighbour $y$ of $x$, then the probability that $y\in A$ is at least $\eta$. In recent joint work with Tim Gowers, we were aiming to obtain a qualitative description of closed subsets of the Cayley graph $\Gamma$ whose vertex set is $\mathbb{F}_2^{n_1}\otimes \dots \otimes \mathbb{F}_2^{n_d}$ with two vertices joined by an edge if their difference is of the form $u_1\otimes \cdots \otimes u_d$. For the matrix case (that is, when $d=2$), such a description was obtained by Khot, Minzer and Safra, a breakthrough that completed the proof of the 2-to-2 conjecture. We have formulated a conjecture for higher dimensions, and proved it in an important special case. In this talk, I will sketch this proof. Also, we have identified a statement about $\eta$-closed sets in Cayley graphs on arbitrary finite Abelian groups that implies the conjecture and can be considered as a "highly asymmetric Balog-Szemerédi-Gowers theorem" when it holds. I will present an example to show that this statement is not true for an arbitrary Cayley graph. It remains to decide whether the statement can be proved for the Cayley graph $\Gamma$.

The last remaining open problem from Erdős and Rényi's original paper on random graphs is the following: for q at least 3, what is the largest d so that the random graph G(n,d/n) is q-colorable with high probability?  A lot of interesting work in probabilistic combinatorics has gone into proving better and better bounds on this q-coloring threshold, but the full answer remains elusive.  However, a non-rigorous method from the statistical physics of glasses - the cavity method - gives a precise prediction for the threshold.  I will give an introduction to the cavity method, with random graph coloring as the running example, and describe recent progress in making parts of the method rigorous, emphasizing the role played by tools from extremal combinatorics.  Based on joint work with Amin Coja-Oghlan, Florent Krzakala, and Lenka Zdeborová.

It is difficult to determine when a graph G can be edge-covered by edge-disjoint copies of a fixed graph F. That is, when it has an F-decomposition. However, if G is large and has a high minimum degree then it has an F-decomposition, as long as some simple divisibility conditions hold. Recent research allows us to prove bounds on the necessary minimum degree by studying a relaxation of this problem, where a fractional decomposition is sought.

I will show how a relatively simple random process can give a good approximation to a fractional decomposition of a dense graph, and how it can then be made exact. This improves the best known bounds for this problem.
 

Consider a family of k-element subsets of an n-element set, and assume that the family does not contain s pairwise disjoint sets. The well-known Erdos Matching Conjecture suggests the maximum size of such a family. Finding the maximum is trivial for n<(s+1)k and is relatively easy for n large in comparison to s,k. There was a splash of activity around the conjecture in the recent years, and, as far as the original question is concerned, the best result is due to Peter Frankl, who verified the conjecture for all n>2sk. In this work, we improve the bound of Frankl for any k and large enough s. We also discuss the connection of the problem to an old question on deviations of sums of random variables going back to the work of Hoeffding and Shrikhande.
 

Numerous problems in extremal hypergraph theory ask to determine the maximal size of a k-uniform hypergraph on n vertices that does not contain an 'enlarged' copy H^+ of a fixed hypergraph H. These include well-known  problems such as the Erdős-Sós 'forbidding one intersection' problem and the Frankl-Füredi 'special simplex' problem.

In this talk we present a general approach to such problems, using a 'junta approximation method' that originates from analysis of Boolean functions. We prove that any (H^+)-free hypergraph is essentially contained in a 'junta' -- a hypergraph determined by a small number of vertices -- that is also (H^+)-free, which effectively reduces the extremal problem to an easier problem on juntas. Using this approach, we obtain, for all C<k<n/C, a complete solution of the extremal problem for a large class of H's, which includes  the aforementioned problems, and solves them for a large new set of parameters.

Based on joint works with David Ellis and Nathan Keller
 

We prove that the dimension of every poset whose comparability graph has maximum degree $\Delta$ is at most $\Delta\log^{1+o(1)} \Delta$. This result improves on a 30-year old bound of F\"uredi and Kahn, and is within a $\log^{o(1)}\Delta$ factor of optimal. We prove this result via the notion of boxicity. The boxicity of a graph $G$ is the minimum integer $d$ such that $G$ is the intersection graph of $d$-dimensional axis-aligned boxes. We prove that every graph with maximum degree $\Delta$ has boxicity at most $\Delta\log^{1+o(1)} \Delta$, which is also within a $\log^{o(1)}\Delta$ factor of optimal. We also show that the maximum boxicity of graphs with Euler genus $g$ is $\Theta(\sqrt{g \log g})$, which solves an open problem of Esperet and Joret and is tight up to a $O(1)$ factor. This is joint work with Alex Scott (arXiv:1804.03271).

We generalise the existence of combinatorial designs to the setting of subset sums in lattices with coordinates indexed by labelled faces of simplicial complexes. This general framework includes the problem of decomposing hypergraphs with extra edge data, such as colours and orders, and so incorporates a wide range of variations on the basic design problem, notably Baranyai-type generalisations, such as resolvable hypergraph designs, large sets of hypergraph designs and decompositions of designs by designs. Our method also gives approximate counting results, which is new for many structures whose existence was previously known, such as high dimensional permutations or Sudoku squares.

We give an alternative, statistical physics based proof of the Ω(d2^{-d}) lower bound for the maximum sphere packing density in dimension d by showing that a random configuration from the hard sphere model has this density in expectation. While the leading constant we achieve is not the best known, we do obtain additional geometric information: we prove a lower bound on the entropy density of sphere packings at this density, a measure of how plentiful such packings are. This is joint work with Felix Joos and Will Perkins.

We characterise the embeddability of simply connected 2-dimensional simplicial complexes in 3-space in a way analogous to Kuratowski’s characterisation of graph planarity, by excluded minors. This answers questions of Lovász, Pardon and Wagner.

 

The Gyárfás-Sumner conjecture says that every graph with huge (enough) chromatic number and bounded clique number contains any given forest as an induced subgraph. The Erdős-Hajnal conjecture says that for every graph H, all graphs not containing H as an induced subgraph have a clique or stable set of polynomial size. This talk is about a third problem related to both of these, the following. Say an n-vertex graph is "c-coherent" if every vertex has degree <cn, and every two disjoint vertex subsets of size at least cn have an edge between them. To prove a given graph H satisfies the Erdős-Hajnal conjecture, it is enough to prove H satisfies the conjecture in all c-coherent graphs and their complements, where c>0 is fixed and as small as we like. But for some graphs H, all c-coherent graphs contain H if c is small enough, so half of the task is done for free. Which graphs H have this property? Paths do (a theorem of Bousquet, Lagoutte, and Thomassé), and non-forests don't. Maybe all forests do? In other words, do all c-coherent graphs with c small enough contain any given forest as an induced subgraph? That question is the topic of the talk. It looks much like the Gyárfás-Sumner conjecture, but it seems easier, and there are already several pretty results. For instance the conjecture is true for all subdivided caterpillars (which is more than we know for Gyárfás-Sumner), and all trees of radius two. Joint work with Maria Chudnovsky, Jacob Fox, Anita Liebenau, Marcin Pilipczuk, Alex Scott and Sophie Spirkl.

A famous theorem of Mantel from 1907 states that every n-vertex graph with more than n^2/4 edges contains at least one triangle. In the 50s, Erdős asked for a quantitative version of this statement: for every n and e, how many triangles must an n-vertex e-edge graph contain?

This question has received a great deal of attention, and a long series of partial results culminated in an asymptotic solution by Razborov, extended to larger cliques by Nikiforov and Reiher. Currently, an exact solution is only known for a small range of edge densities, due to Lovász and Simonovits. In this talk, I will discuss the history of the problem and recent work which gives an exact solution for almost the entire range of edge densities. This is joint work with Hong Liu and Oleg Pikhurko.

The interplay of minimum degree and 'structural properties' of large graphs with a given forbidden subgraph is a central topic in extremal graph theory. For a given graph $F$ we define the homomorphism threshold as the infimum $\alpha$ such that every $n$-vertex $F$-free graph $G$ with minimum degree $>\alpha n$ has a homomorphic image $H$ of bounded size (independent of $n$), which is $F$-free as well. Without the restriction of $H$ being $F$-free we recover the definition of the chromatic threshold, which was determined for every graph $F$ by Allen et al. The homomorphism threshold is less understood and we present recent joint work with O. Ebsen on the homomorphism threshold for odd cycles.

Understanding the distribution of subgraph counts has long been a central question in the study of random graphs. In this talk, we consider the distribution of Sn, the number of K4 subgraphs, in the Erdös Rényi random graph G(n, p). When the edge probability p \in (0, 1) is constant, a classical central limit theorem for Sn states that (Sn−µn)/σn converges in distribution. We establish a stronger form of convergence, namely the corresponding local limit theorem, which is joint work with O. Riordan.
 

A class C of graphs has polynomial expansion if there exists a polynomial p such that for every graph G from C and for every integer r, each minor of G obtained by contracting disjoint subgraphs of radius at most r is p(r)-degenerate. Classes with polynomial expansion exhibit interesting structural, combinatorial, and algorithmic properties. In the talk, I will survey these properties and propose further research directions.

The edge isoperimetric problem for a graph G is to find, for each n, the minimum number of edges leaving any set of n vertices.  Exact solutions are known only in very special cases, for example when G is the usual cubic lattice on Z^d, with edges between pairs of vertices at l_1 distance 1.  The most attractive open problem was to answer this question for the "strong lattice" on Z^d, with edges between pairs of vertices at l_infty distance 1.  Whilst studying this question we in fact solved the edge isoperimetric problem asymptotically for every Cayley graph on Z^d.  I'll talk about how to go from the specification of a lattice to a corresponding near-optimal shape, for both this and the related vertex isoperimetric problem, and sketch the key ideas of the proof. Joint work with Joshua Erde.

Reed conjectured in 1998 that the chromatic number of a graph should be at most the average of the clique number (a trivial lower bound) and maximum degree plus one (a trivial upper bound); in support of this conjecture, Reed proved that the chromatic number is at most some nontrivial convex combination of these two quantities.  King and Reed later showed that a fraction of roughly 1/130000 away from the upper bound holds. Motivated by a paper by Bruhn and Joos, last year Bonamy, Perrett, and I proved that for large enough maximum degree, a fraction of 1/26 away from the upper bound holds. Then using new techniques, Delcourt and I showed that the list-coloring version holds; moreover, we improved the fraction for ordinary coloring to 1/13. Most recently, Kelly and I proved that a 'local' list version holds with a fraction of 1/52 wherein the degrees, list sizes, and clique sizes of vertices are allowed to vary.
 

Aharoni and Berger conjectured that in any bipartite multigraph that is properly edge-coloured by n colours with at least n+1 edges of each colour there must be a matching that uses each colour exactly once (such a matching is called rainbow). This conjecture recently have been proved asymptotically by Pokrovskiy. In this talk, I will consider the same question without the bipartiteness assumption. It turns out that in any multigraph with bounded edge multiplicities, that is properly edge-coloured by n colours with at least n+o(n) edges of each colour, there must be a matching of size n-O(1) that uses each colour at most once. This is joint work with Peter Keevash.

A subset S of initially infected vertices of a graph G is called forcing if we can infect the entire graph by iteratively applying the following process. At each step, any infected vertex which has a unique uninfected neighbour, infects this neighbour. The forcing number of G is the minimum cardinality of a forcing set in G. It was introduced independently as a bound for the minimum rank of a graph, and as a tool in quantum information theory.

The focus of this talk is on the forcing number of the random graph. Furthermore, we will state our bounds on the forcing number of pseudorandom graphs and related problems. The results are joint work with Thomas Kalinowski and Benny Sudakov.

Enumerating families of combinatorial objects with given properties and describing the typical structure of these objects are fundamental problems in extremal combinatorics. In this talk, we will investigate intersecting families of discrete structures in various settings, determining their typical structure as the size of the underlying ground set tends to infinity. Our new approach outlines a general framework for a number of similar problems; in particular, we prove analogous results for hypergraphs, permutations, and vector spaces using the same technique. This is joint work with József Balogh, Shagnik Das, Hong Liu, and Maryam Sharifzadeh.

Given a graph $G$, we can form a hypergraph $H$ whose edges correspond to the triangles in $G$. If $G$ is the standard Erdős-Rényi random graph with independent edges, then $H$ is random, but its edges are not independent, because of overlapping triangles. This is (presumably!) a major complication when proving results about triangles in random graphs.  However, it turns out that, for many purposes, we can treat the triangles as independent, in a one-sided sense (and losing something in the density): we can find an independent random hypergraph within the set of triangles. I will present two proofs, one of which generalizes to larger complete (and some non-complete) subgraphs.

Komlós conjectured in 1981 that among all graphs with minimum degree at least $d$, the complete graph $K_{d+1}$ minimises the number of Hamiltonian subsets, where a subset of vertices is Hamiltonian if it contains a spanning cycle. We prove this conjecture when $d$ is sufficiently large.  In fact we prove a stronger result: for large $d$, any graph $G$ with average degree at least $d$ contains almost twice as many Hamiltonian subsets as $K_{d+1}$, unless $G$ is isomorphic to $K_{d+1}$ or a certain other graph which we specify. This is joint work with Hong Liu, Maryam Sharifzadeh and Katherine Staden.

It is well known that there is a finite colouring of the natural numbers such that there is no infinite set X with X+X (the pairwise sums from X, allowing repetition) monochromatic. It is easy to extend this to the rationals. Hindman, Leader and Strauss showed that there is also such a colouring of the reals, and asked if there exists a space 'large enough' that for every finite colouring there does exist an infinite X with X+X monochromatic. We show that there is indeed such a space. Joint work with Imre Leader.

A central theorem in combinatorics is Sperner’s Theorem, which determines the maximum size of a family in the Boolean lattice that does not contain a 2-chain. Erdos later extended this result and determined the largest family not containing a k-chain. Erdos and Katona and later Kleitman asked how many such chains must appear in families whose size is larger than the corresponding extremal result.

This question was resolved for 2-chains by Kleitman in 1966, who showed that amongst families of size M in the Boolean lattice, the number of 2-chains is minimized by a family whose sets are taken as close to the middle layer as possible. He also conjectured that the same conclusion should hold for all k, not just 2. The best result on this question is due to Das, Gan and Sudakov who showed roughly that Kleitman’s conjecture holds for families whose size is at most the size of the k+1 middle layers of the Boolean lattice. Our main result is that for every fixed k and epsilon, if n is sufficiently large then Kleitman’s conjecture holds for families of size at most (1-epsilon)2^n, thereby establishing Kleitman’s conjecture asymptotically (in a sense). Our proof is based on ideas of Kleitman and Das, Gan and Sudakov.

Joint work with Jozsef Balogh.

Many extremal results on cycles use what may be called BFS method, where a breath first search tree is used as a skeleton to build desired structures. A well-known example is the Bondy-Simonovits theorem that every n-vertex graph with more than 100kn^{1+1/k} edges contains an even cycle of length 2k. The standard BFS method, however, is not easily applicable for supersaturation problems where one wishes to show the existence of many copies of a given  subgraph. The method is also not easily applicable in the hypergraph setting.

In this talk, we focus on some variants of the standard BFS method. We use one of these in conjunction with some useful general reduction theorems that we develop to establish the supersaturation of loose (linear) even cycles in linear hypergraphs. This extends Simonovits' supersaturation theorem on even cycles in graphs. This is joint work with Liana Yepremyan.

If time allows, we will also discuss another variant (joint with Jie Ma) used in the study of Berge cycles of consecutive lengths in hypergraphs.

The $r$-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 percolates.

In this talk I will discuss the proof of a conjecture of Balogh and Bollobás: 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.

The chromatic number of a graph is trivially bounded from above by the maximum degree plus one, and from below by the size of a largest clique. Reed proved in 1998 that compared to the trivial upper bound, we can always save a number of colors proportional to the gap between the maximum degree and the size of a largest clique. A key step in the proof deals with how to spare colors in a graph whose every vertex "sees few edges" in its neighborhood. We improve the existing approach, and discuss its applications to Reed's theorem and strong edge coloring.  This is joint work with Thomas Perrett (Technical University of Denmark) and Luke Postle (University of Waterloo).

Given two $k$-graphs $H$ and $F$, a perfect $F$-packing in $H$ is a collection of vertex-disjoint copies of $F$ in $H$ which together cover all the vertices in $H$. In the case when $F$ is a single edge, a perfect $F$-packing is simply a perfect matching. For a given fixed $F$, it is generally the case that the decision problem whether an $n$-vertex $k$-graph $H$ contains a perfect $F$-packing is NP-complete.

In this talk we describe a general tool which can be used to determine classes of (hyper)graphs for which the corresponding decision problem for perfect $F$-packings is polynomial time solvable. We then give applications of this tool. For example, we give a minimum $\ell$-degree condition for which it is polynomial time solvable to determine whether a $k$-graph satisfying this condition has a perfect matching (partially resolving a conjecture of Keevash, Knox and Mycroft). We also answer a question of Yuster concerning perfect $F$-packings in graphs.

This is joint work with Jie Han (Sao Paulo).
 

We prove tight upper and lower bounds on an observable of the antiferromagnetic Potts model. From this we deduce the case d=3 of a conjecture of Galvin and Tetali on maximising the number of proper colourings in d-regular graphs.

In a recent breakthrough, Peter Keevash proved the Existence conjecture for combinatorial designs, which has its roots in the 19th century. In joint work with Daniela Kühn, Allan Lo and Deryk Osthus, we gave a new proof of this result, based on the method of iterative absorption. In fact, `regularity boosting’ allows us to extend our main decomposition result beyond the quasirandom setting and thus to generalise the results of Keevash. In particular, we obtain a resilience version and a minimum degree version. In this talk, we will present our new results within a brief outline of the history of the Existence conjecture and provide an overview of the proof.

We will consider the following deceptively simple question, formulated recently by Po Shen Loh who connected it to an open problem in Ramsey Theory. Define the '2-less than' relation on the set of triples of integers by saying that a triple x is 2-less than a triple y if x is less than y in at least two coordinates. What is the maximal length of a sequence of triples taking values in {1,...,n} which is totally ordered by the '2-less than' relation?

In his paper, Loh uses the triangle removal lemma to improve slightly on the trivial upper bound of n^2, and conjectures that the truth should be of order n^(3/2). The gap between these bounds has proved to be surprisingly resistant. We shall discuss joint work with Tim Gowers, giving some developments towards this conjecture and a wide array of natural extensions of the problem. Many of these extensions remain open.
 

The Thue-Morse sequence is perhaps the simplest example of an automatic sequence. Various pseudorandomness properties of this sequence have long been studied. During the talk, I will discuss a new result in this direction, asserting that the Gowers uniformity norms of the Thue-Morse sequence are small in a quantitative sense. Similar results hold for the Rudin-Shapiro sequence, as well as for a much wider class of automatic sequences which will be introduced during the talk.

The talk is partially based on joint work with Jakub Byszewski.

Consider the following particle system. We are given a uniform random rooted tree on vertices labelled by $[n] = \{1,2,\ldots,n\}$, with edges directed towards the root. Each node of the tree has space for a single particle (we think of them as cars). A number $m \le n$ of cars arrive one by one, and car $i$ wishes to park at node $S_i$, $1 \le i \le m$, where $S_1, S_2, \ldots, S_m$ are i.i.d. uniform random variables on $[n]$. If a car wishes to park at a space which is already occupied, it follows the unique path oriented towards the root until it encounters an empty space, in which case it parks there; if there is no empty space, it leaves the tree. Let $A_{n,m}$ denote the event that all $m$ cars find spaces in the tree. Lackner and Panholzer proved (via analytic combinatorics methods) that there is a phase transition in this model. Set $m = \lfloor \alpha n \rfloor$. Then if $\alpha \le 1/2$, $\mathbb{P}(A_{n,\lfloor \alpha n \rfloor}) \to \frac{\sqrt{1-2\alpha}}{1-\alpha}$, whereas if $\alpha > 1/2$ we have $\mathbb{P}(A_{n,\lfloor \alpha n \rfloor}) \to 0$. In this talk, we will give a probabilistic explanation for this phenomenon, and an alternative proof via the objective method.

Joint work with Christina Goldschmidt.

The Graham-Pollak theorem states that to decompose the complete graph $K_n$ into complete bipartite subgraphs we need at least $n-1$ of them. What
happens for hypergraphs? In other words, suppose that we wish to decompose the complete $r$-graph on $n$ vertices into complete $r$-partite $r$-graphs; how many do we need?

In this talk we will report on recent progress on this problem. This is joint work with Luka Milicevic and Ta Sheng Tan.

It follows from the ellipsoid method and results of Grotschel, Lovasz and Schrijver that one can find an optimal colouring of a perfect graph in polynomial time. But no ''combinatorial'' algorithm to do this is known.

Here we give a combinatorial algorithm to do this in an n-vertex perfect graph in time O(n^{k+1}^2) where k is the clique number; so polynomial-time for fixed k. The algorithm depends on another result, a polynomial-time algorithm to find a ''balanced skew partition'' in a perfect graph if there is one.

Joint work with Maria Chudnovsky, Aurelie Lagoutte, and Sophie Spirkl.

Given vectors $V = (v_i: i \in [n]) \in R^D$, we define the $V$-intersection of $A,B \subset [n]$ to be the vector $\sum_{i \in A \cap B} v_i$. In this talk, I will discuss a new, essentially optimal, supersaturation theorem for $V$-intersections, which can be roughly stated as saying that any large family of sets contains many pairs $(A,B)$ with $V$-intersection $w$, for a wide range of $V$ and $w$. A famous theorem of Frankl and Rödl corresponds to the case $D=1$ and all $v_i=1$ of our theorem. The case $D=2$ and $v_i=(1,i)$ solves a conjecture of Kalai.

Joint work with Peter Keevash.

The Turán number of an $r$-graph $G$, denoted by $ex(n,G)$, is the maximum number of edges in an $G$-free $r$-graph on $n$ vertices. The Turán density  of an $r$-graph $G$, denoted by $\pi(G)$, is the limit as $n$ tends to infinity of the maximum edge density of an $G$-free $r$-graph on $n$ vertices.

During this talk I will discuss a method, which we call  local stability method, that allows one to obtain exact Turán numbers from Turán density results. This method can be thought of as an extension of the classical stability method by  generically utilising the Lagrangian function. Using it, we obtained new hypergraph Turán numbers. In particular, we did so for a hypergraph called generalized triangle, for uniformities 5 and 6, which solved a conjecture of Frankl and Füredi from 1980's.

This is joint work with Sergey Norin.

For a graph $G$, the $k$-colour Ramsey number $R_k(G)$ is the least integer $N$ such that every $k$-colouring of the edges of the complete graph $K_N$ contains a monochromatic copy of $G$. Let $C_n$ denote the cycle on $n$ vertices. We show that for fixed $k\geq2$ and $n$ odd and sufficiently large,
$$
R_k(C_n)=2^{k-1}(n-1)+1.
$$
This resolves a conjecture of Bondy and Erdős for large $n$. The proof is analytic in nature, the first step of which is to use the regularity method to relate this problem in Ramsey theory to one in nonlinear optimisation.  This allows us to prove a stability-type generalisation of the above and establish a correspondence between extremal $k$-colourings for this problem and perfect matchings in the $k$-dimensional hypercube $Q_k$.

In joint work with Olof Sisask, we establish new quantitative bounds for Roth's theorem on arithmetic progressions, showing that a set of integers with no three-term arithmetic progressions must have density O(1/(log N)^{1+c}) for some absolute constant c>0. This is the integer analogue of a result of Bateman and Katz for the model setting of vector spaces over a finite field, and the proof follows a similar structure. 

The threshold for existence of a giant component in a random graph with given vertex degrees was found by Molloy and Reed (1995), and several authors have since studied the size of the largest and other components in various cases. The critical window was found by Hatami and Molloy (2012), and has a width that depends on whether the asymptotic degree distribution has a finite third moment or not. I will describe some new results (joint work with Remco van der Hofstad and Malwina Luczak) on the barely supercritical case, where this difference between finite and infinite third moment also is seen.

'Erdős-Ko-Rado type problems' are well-studied in extremal combinatorics; they concern the sizes of families of objects in which any two (or any $r$) of the objects in the family 'agree', or 'intersect', in some way.

If $X$ is a finite set, the '$p$-biased measure' on the power-set of $X$ is defined as follows: choose a subset $S$ of $X$ at random by including each element of $X$ independently with probability $p$. If $\mathcal{F}$ is a family of subsets of $X$, one can consider the $p$-biased measure of $\mathcal{F}$, denoted by $\mu_p(\mathcal{F})$, as a function of $p$. If $\mathcal{F}$ is closed under taking supersets, then this function is an increasing function of $p$. Seminal results of Friedgut and Friedgut-Kalai give criteria under which this function has a 'sharp threshold'. Perhaps surprisingly, a careful analysis of the behaviour of this function also yields some rather strong results in extremal combinatorics which do not explicitly mention the $p$-biased measure - in particular, in the field of Erdős-Ko-Rado type problems. We will discuss some of these, including a recent proof of an old conjecture of Frankl that a symmetric three-wise intersecting family of subsets of $\{1,2,\ldots,n\}$ has size $o(2^n)$, and some 'stability' results characterizing the structure of 'large' $t$-intersecting families of $k$-element subsets of $\{1,2,\ldots,n\}$. Based on joint work with (subsets of) Nathan Keller, Noam Lifschitz and Bhargav Narayanan.