Thresholds versus fractional expectation-thresholds

Abstract

Proving a conjecture of Talagrand, a fractional version of the ``expectation-threshold" conjecture of Kalai and the 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 a finite set $X$, where $p_c(\mathcal{F})$ and $q_f(\mathcal{F})$ are the threshold and ``fractional expectation-threshold" of $\mathcal{F}$, and $\ell(\mathcal{F})$ is the maximum size of a minimal member of $\mathcal{F}$. 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--Mézard--Rivoire and Frieze--Sorkin). Our approach builds on a recent breakthrough of Alweiss, Lovett, Wu and Zhang on the Erdős--Rado ``Sunflower Conjecture."