Thresholds and expectation thresholds for larger p

TL;DR

This paper refines a recent result on the relationship between thresholds and expectation thresholds in increasing families of sets, providing a tighter bound that applies to a broader range of probabilities.

Contribution

It improves the Kahn-Kalai conjecture bound by strengthening the inequality relating thresholds and expectation thresholds using a novel approach involving cloned families.

Findings

Strengthened the bound on the threshold $p_c$ in terms of the expectation threshold $q_c$.

Applied the theorem to cloned families to extend the result to larger probabilities.

Reduced the problem to cases where individual element probability is small.

Abstract

Let $p_\mathrm{c}$ and $q_\mathrm{c}$ be the threshold and the expectation threshold, respectively, of an increasing family $\mathcal{F}$ of subsets of a finite set $X$, and let $l$ be the size of a largest minimal element of $\mathcal{F}$. Recently, Park and Pham proved the Kahn-Kalai conjecture, which says that $p_\mathrm{c} \leqslant K q_\mathrm{c} \log_2 l$ for some universal constant $K$. Here we slightly strengthen their result by showing that $p_\mathrm{c} \leqslant 1 - \mathrm{e}^{-K q_\mathrm{c} \log_2 l}$. The idea is to apply the Park-Pham Theorem to an appropriate `cloned' family $\mathcal{F}_k$, reducing the general case (of this and related results) to the case where the individual element probability $p$ is small.