Thresholds, Expectation Thresholds and Cloning - Przybyłowski, Tomasz; Riordan, Oliver (The Electronic Journal of Combinatorics, 2024)
Let p_c and q_c be the threshold and the expectation threshold, respectively, of an increasing family F of subsets of a finite set X, and let l be the size of a largest minimal element of F. Recently, Park and Pham proved the Kahn–Kalai conjecture, which says that p_c K q_c _2 l for some universal constant K. Here, we slightly strengthen their result by showing that p_c 1 - e^-K q_c _2 l. The idea is to apply the Park-Pham Theorem to an appropriate "cloned" family F_k, reducing the general case (of this and related results) to the case where the individual element probability p is small.