Seminar organisers: Jan Grebík and Oleg Pikhurko

Warwick's combinatorics seminar in 2020-21 will be held online 2-3pm UK time on Fridays, occasionally 5:30-6:30pm UK time on Fridays (see the notes below).

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https://us02web.zoom.us/j/82963632253?pwd=Q0psYyt0MURYSy9haWROQzhXRmMwQT09
Meeting ID: 829 6363 2253
Passcode: WC2021

Term 3

Typicality and entropy of processes on infinite trees (Agnes Backhausz), 14:00, ZOOM

When we study random d-regular graphs from the point of view of graph limit theory, the notion of typical processes arise naturally. These are certain invariant families of random variables indexed by the infinite regular tree. Since this tree is the local limit of random d-regular graphs when d is fixed and the number of vertices tends to infinity, we can consider the processes that can be approximated with colorings (labelings) of random d-regular graphs. These are the so-called typical processes, whose properties contain useful information about the structure of finite random regular graphs. In earlier works, various necessary conditions have been given for a process to be typical, by using correlation decay or entropy inequalities. In the work presented in the talk, we go in the other direction and provide sufficient entropy conditions in the special case of edge Markov processes. This condition can be extended to unimodular Galton--Watson random trees as well. Joint work with Charles Bordenave and Balázs Szegedy (https://arxiv.org/abs/2102.02653).

Counting integer partitions with the method of maximum entropy (Gwen McKinley), 17:30, ZOOM, slides

We give an asymptotic formula for the number of partitions of an integer n where the sums of the kth powers of the parts are also fixed, for some collection of values k. To obtain this result, we reframe the counting problem as an optimization problem, and find the probability distribution on the set of all integer partitions with maximum entropy among those that satisfy our restrictions in expectation (in essence, this is an application of Jaynes' principle of maximum entropy). This approach leads to an approximate version of our formula as the solution to a relatively straightforward optimization problem over real-valued functions. To establish more precise asymptotics, we prove a local central limit theorem using an equidistribution result of Green and Tao.

A large portion of the talk will be devoted to outlining how our method can be used to re-derive a classical result of Hardy and Ramanujan, with an emphasis on the intuitions behind the method, and limited technical detail. This is joint work with Marcus Michelen and Will Perkins.

Automorphisms of graphs with a forbidden minor (Pascal Schweitzer), 14:00, ZOOM

According to Frucht’s classic theorem, every finite group is the automorphism group of a finite graph (up to isomorphism). We say the class of all graphs is universal. Babai proved in 1974 that graph classes with a forbidden minor, such as planar graphs, graphs of bounded genus, and graphs of bounded tree width, are not universal. He also made three conjectures regarding automorphisms of such graphs. I will describe a structure theorem of automorphism groups of such graphs, affirming these conjectures.

This is joint work with Martin Grohe and Daniel Wiebking.

Uniform amenability (Gabor Elek), 14:00, ZOOM

According to the classical result of Connes, Feldman and Weiss, measured hyperfiniteness of a group action is equivalent to measured amenability. In the Borel category it is known that hyperfiniteness implies amenability and it is conjectured that the converse is true. Based on the work of Anantharaman-Delaroche and Renault, one can introduce the notion of uniform amenability, a strengthening of measured amenability (it is a sort of exactness in the category of measurable actions, so the famous Gromov-Osajda groups have no free uniformly amenable actions). One can also introduce the notion of uniform hyperfiniteness in a rather natural way. We prove that the two notions are equivalent provided that the measurable action satisfies a boundedness condition for the Radon-Nikodym derivative (e.g. in the case of Poisson boundaries).

Dependency graphs, upper bounds on cumulants and singular graphons (Pierre-Loïc Méliot), 14:00, ZOOM, slides

Consider a sum of random variables S = ∑_{v∈ V} A_v. If the variables A_v are weakly dependent, then it is well known that under mild assumptions, the distribution of S is close to a normal distribution. The theory of dependency graphs enables one to make this statement precise. In this framework, we shall present new bounds on the cumulants of S, which enable one to have a combinatorial approach of this probabilistic results. One of the main application is the study of the fluctuations of the densities of subgraphs in a random graph chosen according to a graphon model. We shall see that two behavior are possible, according to whether the graphon is generic or singular. In the latter case, the limiting distributions that appear are non-Gaussian.

Graphings without measurable perfect matchings (Gabor Kun), 14:00, ZOOM

Measurable matchings of graphings (this is, probability measure preserving Borel graphs) have attracted a great deal of attention in the last decades. The Banach-Tarski paradox shows that the Hall condition does not guarantee the existence of a measurable perfect matching in a bipartite graphing. Laczkovich gave an example when the measurable Hall condition is not sufficient either. We discuss the few such counterexamples.

Regularity method and large deviation principles for the Erdős–Rényi hypergraph (Nicholas Cook), 14:00, ZOOM

In recent years there has been intense activity on the problem of estimating the upper tail for counts of a fixed subgraph in a sparse Erdős–Rényi graph, combining tools and perspectives from diverse areas such as extremal graph theory, statistical physics, stochastic analysis and spectral theory. I will discuss some recent extensions of these results to the hypergraph setting. Our approach rests on new decomposition theorems and counting lemmas for sparse Bernoulli tensors, extending classic results from the regularity method for dense graphs. These results are formulated in terms of a new class of cut-type norms specially tailored for the sparse setting. Based on joint work with Amir Dembo and Huy Tuan Pham.

Towards the sampling Lovász Local Lemma (Vishesh Jain), 14:00, ZOOM

For a constraint satisfaction problem which satisfies the condition of the Lovász local lemma (LLL), the celebrated algorithm of Moser and Tardos allows one to efficiently find a satisfying assignment. In the past few years, much work has gone into understanding whether one can efficiently sample from (approximately) the uniform distribution on satisfying assignments under LLL-like conditions. I will discuss recent progress on this problem, joint with Huy Tuan Pham (Stanford) and Thuy Duong Vuong (Stanford).

A proof of the Erdős–Faber–Lovász conjecture (Tom Kelly), 14:00, ZOOM

The Erdős–Faber–Lovász conjecture (posed in 1972) states that the chromatic index of any linear hypergraph on n vertices is at most n. In joint work with Dong Yeap Kang, Daniela Kühn, Abhishek Methuku, and Deryk Osthus, we proved this conjecture for every sufficiently large n. In this talk, I will present the history of this conjecture and sketch our proof in some special cases.

Polymer models, cluster expansion, and local limit theorems: counting independent sets in the hypercube (Will Perkins), 14:00, ZOOM

I'll discuss how local central limit theorems can be used in combination with two tools from statistical physics, polymer models and the cluster expansion, to asymptotically enumerate independent sets of a given size in the hypercube. Joint work with Matthew Jenssen and Aditya Potukuchi (arxiv:2106.09709)

Term 2

Sparse reconstruction for iid variables (Pál Galicza), 14:00, ZOOM

Term 1

Graph limits, measures, and flows (László Lovász), 14:00, ZOOM

The theory of graph limits is only understood to a somewhat satisfactory degree in the cases of dense graphs and of bounded degree graphs. There is, however, a lot of interest in the intermediate cases. It appears that the most important constituents of graph limits in the general case will be Markov spaces (Markov chains on measurable spaces with a stationary distribution).

This motivates our goal to extend some important theorems from finite graphs to Markov spaces or, more generally, to measurable spaces. In this talk we show that much of flow theory, one of the most important areas in graph theory, can be extended to measurable spaces. In particular, the Hoffman Circulation Theorem, the Max-Flow-Min-Cut Theorem, Multicommodity Flow Theorem, and several other results have simple and elegant extensions to measures.

Graph recolouring (Marthe Bonamy), 14:00, ZOOM

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.

Convergence and limits of finite trees (Gábor Tardos), 14:00, ZOOM, slides

Seeing the success of limit theory of dense finite graphs with graphons as their limit objects we developed a similar (?) theory for finite trees. In order for the sampling limit to make sense we need to make the trees "dense" - we do this by considering them as metric spaces with the normalized graph distance. Using ultaproducts is a simple and elegant way to find unique limit objects (we call them dendrons) and also to highlight similarities and major differences from the theory of dense graph limits. For the underlying quantitative approximation results, one needs more "down to earth" techniques to be developed.
This is joint work with Gábor Elek.

Even-hole free graphs of bounded degree have bounded treewidth (Maria Chudnovsky), 14:00, ZOOM

Tree decompositions are a powerful tool in structural graph theory that is traditionally used in the context of forbidden graph minors. Connecting tree decompositions and forbidden induced subgraphs has so far
largely remained out of reach. Traditionally to bound the treewidth of a graph, one finds a way to decompose it by a so-called laminar collection of decompositions. Recently, in joint work with Tara Abrishami and Kristina
Vuskovic, we proved that even-hole free graphs of bounded degree have bounded tree-width. To do so we used "star cutset separations" that arise naturally in the context of even-hole-free graphs. While the set of star cutset separations is far from being non-crossing, it turns out that one can partition it into a bounded number of laminar collections, and this is sufficient for our purposes.
In this talk we will present an outline of the proof.

Empty axis-parallel boxes (Boris Bukh), 14:00, ZOOM

How to place n points inside the d-dimensional unit cube so every large axis-parallel box contains at least one point? We discuss the motivation as well as a partial solution to this problem. This is a joint work with Ting-Wei Chao.

Mader-perfect digraphs (Tibor Szabó), 14:00, ZOOM

We investigate the relationship of dichromatic number and subdivision containment in digraphs. We call a digraph Mader-perfect if for every (induced) subdigraph F, any digraph of dichromatic number at least v(F) contains an F-subdivision. We show that, among others, arbitrary orientated cycles, bioriented trees, and tournaments on four vertices are Mader-perfect. The first result settles a conjecture of Aboulker, Cohen, Havet, Lochet, Moura, and Thomassé, while the last one extends Dirac's Theorem about 4-chromatic graphs containing a K₄-subdivision to directed graphs. The talk represents joint work with Lior Gishboliner and Raphael Steiner.

Large independent sets from local considerations (Benny Sudakov), 14:00, ZOOM

How well can global properties of a graph be inferred from observations that are purely local? This general question gives rise to numerous interesting problems. One such very natural question was raised independently by Erdos-Hajnal and Linial-Rabinovich in the early 90's. How large must the independence number α(G) of a graph G be whose every m vertices contain an independent set of size r? In this talk we discuss new methods to attack this problem which improve many previous results.

Measurable colorings, the Lovász Local Lemma, and distributed algorithms (Anton Bernshteyn), 14:00, ZOOM

In the past twenty or so years, a rich theory has emerged concerning the behavior of graph colorings, matchings, and other combinatorial notions under additional regularity requirements, for instance measurability. It turns out that this area is closely related to distributed computing, i.e., the part of computer science concerned with problems that can be solved efficiently by a decentralized network of processors. A key role in this relationship is played by the Lovász Local Lemma and its analogs in the measurable setting. In this talk I will outline this relationship and present a number of applications.

Homeomorphs (Bhargav Narayanan), 14:00, ZOOM

Nati Linial asked the following basic problem in 2006: Given a k-dimensional simplicial complex S, how many facets can a k-complex on n vertices have if it contains no topological copy of S? This is a beautiful and natural question, but results in low dimensions apart (k <= 2), very little was previously known. In this talk, I’ll provide an answer in all dimensions and take the scenic route to the answer, surveying many natural problems about simplicial complexes along the way.

Extremal density for sparse minors and subdivisions (Hong Liu), 14:00, ZOOM

How dense a graph has to be to necessarily contain (topological) minors of a given graph H? When H is a complete graph, this question is well understood by result of Kostochka/Thomason for clique minor, and result of Bollobas-Thomason/Komlos-Szemeredi for topological minor. The situation is a lot less clear when H is a sparse graph. We will present a general result on optimal density condition forcing (topological) minors of a wide range of sparse graphs. As corollaries, we resolve several questions of Reed and Wood on embedding sparse minors. Among others,

• (1+o(1))t² average degree is sufficient to force the t x t grid as a topological minor;
• (3/2+o(1))t average degree forces every t-vertex planar graph as a minor, and the constant 3/2 is optimal, furthermore, surprisingly, the value is the same for t-vertex graphs embeddable on any fixed surface;
• a universal bound of (2+o(1))t on average degree forcing every t-vertex graph in any nontrivial minor-closed family as a minor, and the constant 2 is best possible by considering graphs with given treewidth.

Joint work with John Haslegrave and Jaehoon Kim.