Towards an optimal hypergraph container lemma

TL;DR

This paper introduces two new simplified versions of the hypergraph container lemma that improve the understanding of independent sets in hypergraphs, with novel proof techniques and better bounds.

Contribution

It formulates and proves two new versions of the hypergraph container lemma with simpler proofs, alternative notions of almost-independence, and improved bounds on the number of containers.

Findings

Short and simple proofs with connections to probabilistic combinatorics

Use of alternative notions of almost-independence

Improved bounds on the number of containers

Abstract

The hypergraph container lemma is a powerful tool in probabilistic combinatorics that has found many applications since it was first proved a decade ago. Roughly speaking, it asserts that the family of independent sets of every uniform hypergraph can be covered by a small number of almost-independent sets, called containers. In this article, we formulate and prove two new versions of the lemma that display the following three attractive features. First, they both admit short and simple proofs that have surprising connections to other well-studied topics in probabilistic combinatorics. Second, they use alternative notions of almost-independence in order to describe the containers. Third, they yield improved dependence of the number of containers on the uniformity of the hypergraph, hitting a natural barrier for second-moment-type approaches.