A Bayesian Proof of the Spread Lemma

Abstract

A key set-theoretic “spread” lemma has been central to two recent celebrated results in combinatorics: the recentimprovements on the sunflower conjecture by Alweiss, Lovett, Wu, and Zhang; and the proof of the fractionalKahn–Kalai conjecture by Frankston, Kahn, Narayanan, and Park. In this work, we present a new proof of the spreadlemma, that—perhaps surprisingly—takes advantage of an explicit recasting of the proof in the language of Bayesianinference. We show that from this viewpoint the reasoning proceeds in a straightforward and principled probabilisticmanner, leading to a truncated second moment calculation which concludes the proof.