The Park-Pham Theorem with Optimal Convergence Rate

TL;DR

This paper improves the Park-Pham Theorem by establishing an optimal convergence rate with respect to the parameter epsilon, refining bounds related to graph and hypergraph thresholds.

Contribution

It provides a version of the Park-Pham Theorem with the best possible epsilon dependence, advancing understanding of probabilistic thresholds in combinatorics.

Findings

Achieves optimal epsilon dependence in the theorem

Refines bounds on probabilistic thresholds

Enhances theoretical understanding of graph properties

Abstract

Park and Pham's recent proof of the Kahn-Kalai conjecture was a major breakthrough in the field of graph and hypergraph thresholds. Their result gives an upper bound on the threshold at which a probabilistic construction has a $1-\epsilon$ chance of achieving a given monotone property. While their bound in other parameters is optimal up to constant factors for any fixed $\epsilon$, it does not have the optimal dependence on $\epsilon$ as $\epsilon\rightarrow 0$. In this short paper, we prove a version of the Park-Pham Theorem with optimal $\epsilon$-dependence.