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arXiv · 2203.17207

A Proof of the Kahn-Kalai Conjecture

Jinyoung Park·Huy Tuan Pham
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Abstract

Proving the ``expectation-threshold'' conjecture of Kahn and Kalai, 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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Jinyoung Park, Huy Tuan Pham. 2023-04-13. A Proof of the Kahn-Kalai Conjecture. https://arxiv.org/abs/2203.17207

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