On the probability of nonexistence in binomial subsets

TL;DR

This paper develops precise asymptotic estimates for the probability that a random subset of vertices in a hypergraph is independent, using cumulant-based bounds, with applications to random graphs and number theory.

Contribution

It introduces a novel cumulant-based framework for asymptotic probability estimates of independence in random hypergraph subsets, unifying and extending prior methods.

Findings

Provides explicit bounds for independence probabilities

Asymptotic bounds coincide under natural conditions

Applicable to random graphs and arithmetic progressions

Abstract

Given a hypergraph $\Gamma=(\Omega,\mathcal{X})$ and a sequence $\mathbf{p} = (p_\omega)_{\omega\in \Omega}$ of values in $(0,1)$, let $\Omega_{\mathbf{p}}$ be the random subset of $\Omega$ obtained by keeping every vertex $\omega$ independently with probability $p_\omega$. We investigate the general question of deriving fine (asymptotic) estimates for the probability that $\Omega_{\mathbf{p}}$ is an independent set in $\Gamma$, which is an omnipresent problem in probabilistic combinatorics. Our main result provides a sequence of upper and lower bounds on this probability, each of which can be evaluated explicitly in terms of the joint cumulants of small sets of edge indicator random variables. Under certain natural conditions, these upper and lower bounds coincide asymptotically, thus giving the precise asymptotics of the probability in question. We demonstrate the applicability of our results with two concrete examples: subgraph containment in random (hyper)graphs and arithmetic progressions in random subsets of the integers.