Probabilistic hypergraph containers

TL;DR

This paper introduces a probabilistic approach to hypergraph container theorems, showing that large random subsets are likely to be nearly independent or contain almost-independent structures, strengthening existing inequalities.

Contribution

It provides a probabilistic variant of hypergraph container theorems that is simpler and effective for many applications, especially in the range where traditional inequalities are weak.

Findings

Large random subsets are almost independent with high probability.

The method strengthens Janson's inequality for large subset sizes.

It offers a probabilistic alternative to deterministic container theorems.

Abstract

Given a $k$-uniform hypergraph $\mathcal{H}$ and sufficiently large $m \gg m_0(\mathcal{H})$, we show that an $m$-element set $I \subseteq V(\mathcal{H})$, chosen uniformly at random, with probability $1 - e^{-\omega(m)}$ is either not independent or is contained in an almost-independent set in $\mathcal{H}$ which, crucially, can be constructed from carefully chosen $o(m)$ vertices of $I$. As a corollary, this implies that if the largest almost-independent set in $\mathcal{H}$ is of size $o(v(\mathcal{H}))$ then $I$ itself is an independent set with probability $e^{-\omega(m)}$. More generally, $I$ is very likely to inherit structural properties of almost-independent sets in $\mathcal{H}$. The value $m_0(\mathcal{H})$ coincides with that for which Janson's inequality gives that $I$ is independent with probability at most $e^{-\Theta(m_0)}$. On the one hand, our result is a significant strengthening of Janson's inequality in the range $m \gg m_0$. On the other hand, it can be seen as a probabilistic variant of hypergraph container theorems, developed by Balogh, Morris and Samotij and, independently, by Saxton and Thomason. While being strictly weaker than the original container theorems in the sense that it does not apply to all independent sets of size $m$, it is nonetheless sufficient for many applications and admits a short proof using probabilistic ideas.