  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.
