Plenary talks (titles and abstracts)

Julia Böttcher: The size Ramsey number of degenerate graphs and partition universality of G(n,p)
The r-colour size-Ramsey number of a graph G is the minimum number of edges of a graph H such that any r-colouring of the edges of H has a monochromatic G-copy.
Random graphs play an important role in the study of size-Ramsey numbers. Using random graphs, we establish a new bound on the size-Ramsey number of D-degenerate graphs with bounded maximum degree.

In the talk I will summarise what is known about size-Ramsey numbers, explain the connection to random graphs and their so-called partition universality, and outline which methods we use in our proof.

Based on joint work with Peter Allen.

Boris Bukh: Random algebraic constructions
I will present a class of random constructions that rely on random polynomials. These constructions have a pleasant property that the good events happen almost simultaneously. I will then show how these constructions can be used to construct near-optimal extremal graphs, and related objects.

Peter Keevash: Colouring tori
We prove conjectures of Engbers and Galvin and of Kahn and Park on the number of q-colourings of a discrete torus Z_m^d (we give asymptotics in d for fixed q and even m). Our methods also apply to homomorphisms. This is joint work with Matthew Jenssen.

Andrzej Ruciński: Embedding Erdős-Rényi Random Graphs into Random Regular Graphs
The relationship between Erdős-Rényi models \(G(n,m)\) and regular models \(\G(n,d)\) of random graphs was rst studied by Kim and Vu in 2004. In particular, they showed that in a wide range of \(m\), one can couple the two so that a.a.s. \(\G(n,m)\subset \G(n,d)\), where \(m=(1-o(1))\tfrac{2m}n\). Later Dudek, Frieze, Ruciński, and Šileikis extended the range of m a little and also generalized their result to random \(k\)-uniform hypergraphs. Still, the range of \(m\) has been limited to \(m=o(n^k)\).
Recently, while proving an analogous result for bipartite random graphs we found a way to cover the case \(m=\Theta(n^2)\) as well. Besides standard switching techniques, the proof relies heavily on a purely deterministic result about the presence of alternating cycles in two-colored quasi-random graphs. During my talk I will outline the proof in the bipartite model. This is joint work with T. Klimošová, Chr. Reiher, and M. Šileikis.

Alex Scott: Graphs with forbidden induced subgraphs
It is well known (for example by considering random graphs) that a graph on n vertices need not have complete subgraphs or independent sets of size more than about log n. But what if we consider graphs which do not contain some specific induced subgraph? Erdos and Hajnal conjectured in the 1980s that for every graph H there is a constant c such that every graph on n vertices without an induced copy of H contains a clique or stable set of size n^c. The Erdos-Hajnal conjecture is still very much open, but we will discuss some progress and related results. This includes joint work with Maria Chudnovsky, Jacob Fox, Paul Seymour and Sophie Spirkl.

Asaf Shapira: Testing graphs against an unknown distribution
The area of graph property testing seeks to understand the relation between the global properties of a graph and its local statistics. In the classical model, the local statistics of a graph is defined relative to a uniform distribution over the graph's vertex set. A graph property P is said to be testable if the local statistics of a graph can allow one to distinguish between graphs satisfying P and those that are far from satisfying it.
Goldreich recently introduced a generalization of this model in which one endows the vertex set of the input graph with an arbitrary and unknown distribution, and asked which of the properties that can be tested in the classical model can also be tested in this more general setting. We completely resolve this problem by giving a (surprisingly "clean'') characterization of these properties. To this end we prove a removal lemma for vertex weighted graphs which is of independent interest.

Joint work with L. Gishboliner

Dan Spielman: Finite Free Probability and Ramanujan Graphs
We introduce a finite analog of Voiculescu's Free Probability that allows us to compute the expected characteristic polynomials of certain random matrices and to prove bounds on the locations of the roots of those polynomials. We sketch how this theory may be used to prove that there exist bipartite Ramanujan graphs of every degree and number of vertices.

No prior knowledge of free probability or Ramanujan graphs will be assumed.