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

Abstract: We show that the uniform measure on copies of a graph $H$ is $Cq_H\log(2e(H))$-spread, where $q_H$ is its graphic expectation threshold defined using expected count one. This gives a fractional expectation threshold of at most $C\pe(H)\log(2e(H))$. We remove the logarithmic loss for trees and for graphs whose maximum degree is at most exponential in their average degree. The ``second'' Kahn-Kalai conjecture therefore holds for all such graphs.