Constant-factor comparison of fractional and expectation thresholds

Establish a universal constant-factor comparison between the fractional expectation threshold and the expectation threshold: prove that there exists a universal constant C such that every nontrivial increasing family F on every finite ground set V satisfies q_f(F) ≤ C q(F), equivalently, prove that every weakly p-small increasing family is (p/C)-small.

Background

For an increasing family F of subsets of a finite ground set V, the expectation threshold q(F) is defined through p-smallness, while the fractional expectation threshold q_f(F) is defined through weak p-smallness. Since every p-small family is weakly p-small, one always has q(F) ≤ q_f(F).

Before this paper, the best general comparison depended on the ground-set size through an O(log log(N+2)) factor, and a dimension-free result incurred a loss depending on q(F). The paper proves the improved bound q_f(F) ≤ 64 log*_2(N+2) q(F), but this still does not establish the conjectured universal constant-factor comparison. The conjecture is therefore the strongest unresolved form of the integral-to-fractional-cover comparison discussed in the paper.