Chv\'{a}tal's Conjecture and Correlation Inequalities

TL;DR

This paper explores Chvátal's conjecture in extremal combinatorics by reformulating it through influences on Boolean functions and correlation inequalities, analyzing special cases with discrete Fourier analysis.

Contribution

It introduces a new reformulation of Chvátal's conjecture using Boolean influences and correlation inequalities, and investigates specific cases with Fourier analysis techniques.

Findings

Reformulation of Chvátal's conjecture via Boolean influences

Analysis of special cases using discrete Fourier analysis

Insights into correlation inequalities related to the conjecture

Abstract

Chv\'{a}tal's conjecture in extremal combinatorics asserts that for any decreasing family $\mathcal{F}$ of subsets of a finite set $S$, there is a largest intersecting subfamily of $\mathcal{F}$ consisting of all members of $\mathcal{F}$ that include a particular $x \in S$. In this paper we reformulate the conjecture in terms of influences of variables on Boolean functions and correlation inequalities, and study special cases and variants using tools from discrete Fourier analysis.