The Park-Pham Theorem with Optimal Convergence Rate

10/07/2022

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by   Tolson Bell, et al.

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Park and Pham's recent proof of the Kahn–Kalai conjecture was a major breakthrough in the field of graph and hypergraph thresholds. Their result gives an upper bound on the threshold at which a probabilistic construction has a 1-ϵ chance of achieving a given monotone property. While their bound in other parameters is optimal up to constant factors for any fixed ϵ, it does not have the optimal dependence on ϵ as ϵ→ 0. In this short paper, we prove a version of the Park–Pham Theorem with optimal ϵ-dependence.