Thresholds and expectation thresholds for larger p

Let $p_{\mathrm{c}}$ and $q_{\mathrm{c}}$ be the threshold and the expectation threshold, respectively, of an increasing family $\mathcal{F}$ of subsets of a finite set $X$, and let $l$ be the size of a largest minimal element of $\mathcal{F}$. Recently, Park and Pham proved the Kahn-Kalai conjecture, which says that $p_{\mathrm{c}}⩽K{q}_{\mathrm{c}}{\log }_{2}l$ for some universal constant $K$. Here we slightly strengthen their result by showing that $p_{\mathrm{c}}⩽1-{\mathrm{e}}^{-K{q}_{\mathrm{c}}{\log }_{2}l}$. The idea is to apply the Park-Pham Theorem to an appropriate `cloned’ family ${\mathcal{F}}_k$, reducing the general case (of this and related results) to the case where the individual element probability $p$ is small.