A Proof of the Kahn-Kalai Conjecture

Proving the “expectation-threshold” conjecture of Kahn and Kalai, we show that for any increasing property $\mathcal{F}$ on a finite set X,equationp_c(F)=O(q(F)(F)),equationwhere $p_{c}(\mathcal{F})$ and $q(\mathcal{F})$ are the threshold and “expectation threshold” of $\mathcal{F}$, and $\ell(\mathcal{F})$ is the maximum size of a minimal member of $\mathcal{F}$.