A proof of the Kahn–Kalai conjecture

Proving the "expectation-threshold" conjecture of Kahn and Kalai [Combin. Probab. Comput. 16 (2007), pp. 495–502], we show that for any increasing property F \mathcal {F} on a finite set X X , \[ p c ( F ) = O ( q ( F ) log ⁡ ℓ ( F ) ) , p_c(\mathcal {F})=O(q(\mathcal {F})\log \ell (\mathcal {F})), \] where p c ( F ) p_c(\mathcal {F}) and q ( F ) q(\mathcal {F}) are the threshold and "expectation threshold" of F \mathcal {F} , and ℓ ( F ) \ell (\mathcal {F}) is the maximum of 2 2 and the maximum size of a minimal member of F \mathcal {F} .