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 $p_\mathrm{ c} \leqslant K q_\mathrm{ c} \log_2 l$ for some universal constant $K$. Here we slightly strengthen their result by showing that $p_\mathrm{ c} \leqslant 1 - \mathrm{ e}^{ -K q_\mathrm{ c} \log_2 l}$. 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.
Categories: math.CO math.PR