A Log-Star Comparison Between Expectation Threshold and Fractional Expectation Threshold

Abstract: We proved a log-star comparison between the expectation threshold $q(\mathcal F)$ and the fractional expectation threshold $q_f(\mathcal F)$ for any nontrivial increasing family $\mathcal F$ on a finite ground set $V$ of size $|V|=N$. Specifically, we show that $$q_f(\mathcal F)\le64\log_2^{*(N+2)\,q(\mathcal} F),$$ where we define $\log_2^* x$ to be the least integer $k\ge0$ such that applying $\log_2$ repeatedly $k$ times gives a number at most $1$.