The Park-Pham Theorem with Optimal Convergence Rate

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