A dimension-free comparison between expectation thresholds and fractional expectation thresholds

We prove a dimension-free 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. Specifically, we show that there is a universal constant $K>0$ such that $$q_f(\mathcal{F})\le Kq(\mathcal{F})\max \{1,\log \log (1\mathrm{/}q(\mathcal{F}))\}\mathrm{.}$$ Combining this comparison with a recent result of Li [arXiv:2609.08967] on the fractional version of Talagrand’s discrete Convexity Conjecture, we obtain a dimension-independent bound toward the conjecture.