A short proof of Kahn-Kalai conjecture

Abstract

In a recent paper, Park and Pham famously proved Kahn-Kalai conjecture. In this note, we simplify their proof, using an induction to replace the original analysis. This reduces the proof to one page, and from the argument it is also easy to read that one can set the constant $K$ in the conjecture to $\approx 3.998$, which could be the best value under the current method. Our argument also applies to the $\epsilon$-version of Park-Pham result, studied by Bell.