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 that (Formula Presented) for any increasing family F on a finite set X, where p_c(F) and q_f (F) are the threshold and "fractional expectation-threshold" of F, and l(F) is the maximum 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 graphs (new). We also resolve (and vastly extend) the "axial" version of the random multi-dimensional assignment problem (earlier considered by Martin-Mezard-Rivoire and Frieze{Sorkin). Our approach builds on a recent breakthrough of Alweiss, Lovett, Wu and Zhang on the Erdos-Rado "Sun ower Conjecture."