In this survey we aim to give a comprehensive overview of results using sublinear expanders. The term sublinear expanders refers to a variety of definitions of expanders, which typically are defined to be graphs G such that every not-too-small and not-too-large set of vertices U has neighbourhood of size at least α|U|, where α is a function of n and |U|. This is in contrast with linear expanders, where α is typically a constant. We will briefly describe proof ideas of some of the results mentioned here, as well as related open problems.

A Latin square is an n by n grid filled with n symbols so that each symbol appears exactly once in each row and each column. A transversal in a Latin square is a collection of cells which do not share any row, column, or symbol. This survey will focus on results from the last decade which have continued the long history of the study of transversals in Latin squares.

Introduced by Erdős in 1950, a covering system of the integers is a finite collection of infinite arithmetic progressions whose union is the set of all integers. Many beautiful questions and conjectures about covering systems have been posed over the past several decades, but until the last decade little was known about their properties. Most famously, the so-called minimum modulus problem of Erdős was resolved in 2015 by Hough, who proved that in every covering system with distinct moduli, the minimum modulus is at most a fixed constant. The ideas of Hough were simplified and extended in 2018 by Balister, Bollobas, Morris, Sahasrabudhe and Tiba, to give solutions (or progress towards solutions) to a number of related questions. We give a summary of this and other progress that has been made since.

Which conditions ensure that a digraph contains all oriented paths of some given length, or even a all oriented trees of some given size, as a subgraph? One possible condition could be that the host digraph is a tournament of a certain order. In arbitrary digraphs and oriented graphs, conditions on the chromatic number, on the edge density, on the minimum outdegree and on the minimum semidegree have been proposed. In this survey, we review the known results, and highlight some open questions in the area.

The cluster expansion is a classical tool from statistical physics used to study the phase diagram of interacting particle systems. Recently, the cluster expansion has seen a number of applications in combinatorics and the field of approximate counting/sampling. In this article, we give an introduction to the cluster expansion and survey some of these recent developments.

The slice rank polynomial method was introduced by Tao in 2016 following the breakthrough of Ellenberg and Gijswijt on the famous Cap-Set Problem, which in turn was building on work of Croot, Lev and Pach. This survey gives an introduction to the slice rank polynomial method, shows some of its early applications, and discusses the developments since then.

Chow rings of toric varieties, which originate in intersection theory, feature a rich combinatorial structure of independent interest. We survey four different ways of computing in these rings, due to Billera, Brion, Fulton–Sturmfels, and Allermann–Rau. We illustrate the beauty and power of these methods by giving four proofs of Huh and Huh–Katz’s formula μk(Μ) = degΜ(αr–k βk) for the coefficients of the reduced characteristic polynomial of a matroid M as the mixed intersection numbers of the hyperplane and reciprocal hyperplane classes α and β in the Chow ring of Μ. Each of these proofs sheds light on a different aspect of matroid combinatorics, and provides a framework for further developments in the intersection theory of matroids.

Our presentation is combinatorial, and does not assume previous knowledge of toric varieties, Chow rings, or intersection theory. This survey was prepared for the Clay Lecture to be delivered at the 2024 British Combinatorics Conference.

About twenty years ago, Green wrote a survey article on the utility of looking at toy versions over finite fields of problems in additive combinatorics. This survey was extremely influential, and the rapid development of additive combinatorics necessitated a follow-up article ten years later, which was written by Wolf. Since the publication of Wolf’s article, an immense amount of progress has been made on several central open problems in additive combinatorics in both the finite field model and integer settings. This survey covers some of the most significant results of the past ten years and suggests future directions.

Network flows over time are a fascinating generalization of classical (static) network flows, introducing an element of time. They naturally model problems where travel and transmission are not instantaneous and flow may vary over time. Not surprisingly, flow over time problems turn out to be more challenging to solve than their static counterparts. In this survey, we mainly focus on the efficient computation of transshipments over time in networks with several source and sink nodes with given supplies and demands, which is arguably the most difficult flow over time problem that can still be solved in polynomial time.