Thresholds versus fractional expectation-thresholds

Proving a conjecture of Talagrand, a fractional version of the 'expectation-threshold' conjecture of Kalai and the second author, we show for any increasing family $F$ on a finite set $X$ that $p_c (F) =O( q_f (F) \log \ell(F))$, where $p_c(F)$ and $q_f(F)$ are the threshold and 'fractional expectation-threshold' of $F$, and $\ell(F)$ is the largest size of a minimal member of $F$. This easily implies several heretofore difficult results and conjectures in probabilistic combinatorics, including thresholds for perfect hypergraph matchings (Johansson--Kahn--Vu), bounded-degree spanning trees (Montgomery), and bounded-degree spanning graphs (new). We also resolve (and vastly extend) the 'axial' version of the random multi-dimensional assignment problem (earlier considered by Martin--M\'{e}zard--Rivoire and Frieze--Sorkin). Our approach builds on a recent breakthrough of Alweiss, Lovett, Wu and Zhang on the Erd\H{o}s--Rado 'Sunflower Conjecture'.