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 p(c) (F) = O(q(f) (T) log l(F)) 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 (Mont-gomery), 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 Sunflower Conjecture.

Automated summary

This research validates Talagrand's conjecture and proves some challenging results and conjectures in probabilistic combinatorics, including perfect hypergraph matchings and bounded degree spanning trees.