Fractional expectation thresholds and the"second"Kahn-Kalai conjecture

Abstract

We show that the uniform probability measure on copies of a nonempty graph $H$ in $K_n$ is $Cq_H\log(2e(H))$-spread, where $q_H$ is its graphic expectation threshold. Consequently, the fractional expectation threshold of $H$ is at most $Cq_H\log(2e(H))$. We remove the logarithmic factor for trees and for graphs whose average degree is at least the logarithm of their maximum degree. This proves the ``second''Kahn-Kalai conjecture for these two classes, which encompass most of the standard families studied in random graph containment problems.