Some Results on Fractional vs. Expectation Thresholds

Abstract

A conjecture of Talagrand (2010) states that the so-called expectation and fractional expectation thresholds are always within at most some constant factor from each other. The expectation threshold $q$ for an increasing class $\mathcal{F}\subseteq 2^X$ allows to locate the threshold for $\mathcal{F}$ within a logarithmic factor. The same holds for the fractional expectation threshold $q_f$. These are important breakthrough results of Park and Pham (2022), resp. Frankston, Kahn, Narayanan and Park (2019). We will survey what is known about the relation between $q$ and $q_f$ and prove some further special cases of Talagrand’s conjecture.