Hypergraph containers

Abstract

We develop a notion of containment for independent sets in hypergraphs. For every $r$-uniform hypergraph $G$, we find a relatively small collection $C$ of vertex subsets, such that every independent set of $G$ is contained within a member of $C$, and no member of $C$ is large; the collection, which is in various respects optimal, reveals an underlying structure to the independent sets. The containers offer a straightforward and unified approach to many combinatorial questions concerned (usually implicitly) with independence. With regard to colouring, it follows that simple $r$-uniform hypergraphs of average degree $d$ have list chromatic number at least $(1/(r-1)^2 + o(1)) \log_r d$. For $r = 2$ this improves a bound due to Alon and is tight. For $r \ge 3$, previous bounds were weak but the present inequality is close to optimal. In the context of extremal graph theory, it follows that, for each $\ell$-uniform hypergraph $H$ of order $k$, there is a collection $C$ of $\ell$-uniform hypergraphs of order $n$ each with $o(n^k)$ copies of $H$, such that every $H$-free $\ell$-uniform hypergraph of order $n$ is a subgraph of a hypergraph in $C$, and $\log |C| \le c n^{\ell-1/m(H)} \log n$ where $m(H)$ is a standard parameter (there is a similar statement for induced subgraphs). This yields simple proofs, for example, for the number of $H$-free hypergraphs, and for the sparsity theorems of Conlon-Gowers and Schacht. A slight variant yields a counting version of the K{\L}R conjecture. Likewise, for systems of linear equations the containers supply, for example, bounds on the number of solution-free sets, and the existence of solutions in sparse random subsets. Balogh, Morris and Samotij have independently obtained related results.