Thresholds versus fractional expectation-thresholds

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