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

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, $\ell\geq 1$.