On Sharp Thresholds of Monotone Properties: Bourgain's Proof Revisited

TL;DR

This paper revisits Bourgain's proof of a theorem on sharp thresholds of monotone properties, providing detailed explanations and updated notation to clarify the original argument.

Contribution

It offers an expository reproof of Bourgain's theorem with additional details and modernized notation, enhancing understanding of sharp threshold phenomena.

Findings

Clarifies Bourgain's proof with detailed exposition

Updates notation for better clarity

Strengthens understanding of sharp threshold behavior

Abstract

The purpose of this expository note is to give the proof of a theorem of Bourgain with some additional details and updated notation. The theorem first appeared as an appendix to the breakthrough paper by Friedgut, \emph{Sharp Thresholds of graph properties and the $k$-SAT Problem}. Throughout, we use notation and definitions akin to those in O'Donnell's book, \emph{Analysis of Boolean Functions}.