A second moment proof of the spread lemma

TL;DR

This paper presents a new, probabilistic proof of the spread lemma using Bayesian inference, simplifying previous counting and entropy-based proofs, and highlighting connections to the planting trick in random CSPs.

Contribution

It introduces a second moment proof of the spread lemma based on Bayesian inference, offering a more straightforward and principled approach.

Findings

The proof offers a clear probabilistic perspective on the spread lemma.

It demonstrates the effectiveness of Bayesian inference in combinatorial proofs.

Connections to the planting trick in random CSPs are elucidated.

Abstract

This note concerns a well-known result which we term the ``spread lemma,'' which establishes the existence (with high probability) of a desired structure in a random set. The spread lemma was central to two recent celebrated results: (a) the improved bounds of Alweiss, Lovett, Wu, and Zhang (2019) on the Erd\H{o}s-Rado sunflower conjecture; and (b) the proof of the fractional Kahn--Kalai conjecture by Frankston, Kahn, Narayanan and Park (2019). While the lemma was first proved (and later refined) by delicate counting arguments, alternative proofs have also been given, via Shannon's noiseless coding theorem (Rao, 2019), and also via manipulations of Shannon entropy bounds (Tao, 2020). In this note we present a new proof of the spread lemma, that takes advantage of an explicit recasting of the proof in the language of Bayesian statistical inference. We show that from this viewpoint the proof proceeds in a straightforward and principled probabilistic manner, leading to a truncated second moment calculation which concludes the proof. The proof can also be viewed as a demonstration of the ``planting trick'' introduced by Achlioptas and Coga-Oghlan (2008) in the study of random constraint satisfaction problems.