A simple proof of the non-uniform Kahn-Kalai conjecture

TL;DR

This paper offers a simplified proof of the non-uniform Kahn-Kalai conjecture, extending its applicability and providing near-optimal bounds for various set systems and sampling probabilities.

Contribution

It presents a reformulated, simpler proof that applies to non-uniform measures and yields improved bounds for the conjecture.

Findings

Applicable to non-uniform product measures

Provides near-optimal bounds for high sampling probabilities

Establishes a clean bound for -bounded set systems

Abstract

We revisit the Kahn-Kalai conjecture, recently proved in striking fashion by Park and Pham, and present a slightly reformulated simple proof which has a few advantages: (1) it works for non-uniform product measures, (2) it gives near-optimal bounds even for sampling probabilities close to 1, (3) it gives a clean bound of $p_c \leq 4q_c \log_2 (7\ell)$ for every $\ell$-bounded set system, $\ell\geq 1$.