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 $1-\epsilon$ chance of achieving a given monotone property. While their bound in other parameters is optimal up to constant factors for any fixed $\epsilon$, it does not have the optimal dependence on $\epsilon$ as $\epsilon\rightarrow 0$. In this short paper, we prove a version of the Park-Pham Theorem with optimal $\epsilon$-dependence.

翻译：Park与Pham近期对Kahn-Kalai猜想的证明是图与超图阈值领域的重大突破。该结果给出了概率构造以$1-\epsilon$概率实现给定单调性质的阈值上界。尽管对于任意固定$\epsilon$，其参数余项在常数因子意义下已达最优，但该界限在$\epsilon\rightarrow 0$时的$\epsilon$依赖性并非最优。本文通过简短证明，给出了具有最优$\epsilon$依赖性的Park-Pham定理版本。