Supersaturation for hereditary properties

TL;DR

This paper establishes a supersaturation theorem for hereditary properties in random hypergraphs, showing that the probability of having fewer than a certain number of induced subgraphs from a hereditary family is exponentially small.

Contribution

It extends the supersaturation concept to hereditary properties in random hypergraphs, providing bounds on the probability of fewer induced subgraphs than expected.

Findings

Probability of fewer than δn^t induced subgraphs is exponentially small.

Generalizes supersaturation to hereditary properties in hypergraphs.

Answers a question posed by Bollobás and Nikiforov.

Abstract

Let $\mathcal{F}$ be a collection of $r$-uniform hypergraphs, and let $0 < p < 1$. It is known that there exists $c = c(p,\mathcal{F})$ such that the probability of a random $r$-graph in $G(n,p)$ not containing an induced subgraph from $\mathcal{F}$ is $2^{(-c+o(1)){n \choose r}}$. Let each graph in $\mathcal{F}$ have at least $t$ vertices. We show that in fact for every $\epsilon > 0$, there exists $\delta = \delta (\epsilon, p,\mathcal{F}) > 0$ such that the probability of a random $r$-graph in $G(n,p)$ containing less than $\delta n^t$ induced subgraphs each lying in $\mathcal{F}$ is at most $2^{(-c+\epsilon){n \choose r}}$. This statement is an analogue for hereditary properties of the supersaturation theorem of Erd\H{o}s and Simonovits. In our applications we answer a question of Bollob\'as and Nikiforov.