A proof of the Kahn–Kalai conjecture

Park, Jinyoung; Pham, Huy

doi:10.1090/jams/1028

A proof of the Kahn–Kalai conjecture
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by Jinyoung Park and Huy Tuan Pham

J. Amer. Math. Soc. 37 (2024), 235-243

DOI: https://doi.org/10.1090/jams/1028

Published electronically: August 7, 2023

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Abstract:

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

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