Thresholds versus Fractional Expectation-thresholds

We discuss our result on a conjecture of Talagrand, which is a fractional version of the “expectation-threshold” conjecture of Kahn and Kalai. This easily implies various difficult results in probabilistic combinatorics, e.g. thresholds for perfect hypergraph matchings (Johansson-Kahn-Vu) and bounded-degree spanning trees (Montgomery). Our approach builds on recent breakthrough work of Alweiss, Lovett, Wu, and Zhang on the Erdős-Rado “Sunflower Conjecture.” 

This is joint work with Keith Frankston, Jeff Kahn, and Bhargav Narayanan.