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

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 $p_c\le 4q_c{\log }_2(7\ell )$ for every $\ell$-bounded set system, $\ell \ge 1$.