The random k-matching-free process

Let $\mathcal{P}$ be a graph property which is preserved by removal of edges, and consider the random graph process that starts with the empty $n$-vertex graph and then adds edges one-by-one, each chosen uniformly at random subject to the constraint that $\mathcal{P}$ is not violated. These types of random processes have been the subject of extensive research over the last 20 years, having striking applications in extremal combinatorics, and leading to the discovery of important probabilistic tools. In this paper we consider the $k$-matching-free process, where $\mathcal{P}$ is the property of not containing a matching of size $k$. We are able to analyse the behaviour of this process for a wide range of values of $k$; in particular we prove that if $k=o(n)$ or if $n-2k=o(\sqrt{n}/\log n)$ then this process is likely to terminate in a $k$-matching-free graph with the maximum possible number of edges, as characterised by Erd\H{o}s and Gallai. We also show that these bounds on $k$ are essentially best possible, and we make a first step towards understanding the behaviour of the process in the intermediate regime

Tiling Dense Hypergraphs

There are three essentially necessary conditions for perfect tilings in hypergraphs, which correspond to barriers in space, divisibility and covering. It is natural to ask when these conditions are asymptotically sufficient. Our main result confirms this for hypergraph families that are approximately closed under taking a typical induced subgraph of constant order. As an application, we recover and extend a series of well-known results for perfect tilings in hypergraphs and related settings involving vertex orderings and rainbow structures