A Correlation Inequality on Three Functions

TL;DR

This paper investigates a correlation inequality involving three upward closed set systems in a Boolean lattice, challenging a conjecture about the maximum density of points in exactly one of three such sets.

Contribution

It provides a negative answer to Kahn's question, showing the set of points in exactly one of three upward closed systems can exceed the conjectured density bound.

Findings

Counterexample disproves the conjecture.

The density of points in exactly one of three systems can be greater than 4/9.

The result extends understanding of correlation inequalities in combinatorics.

Abstract

Let $X$ and $Y$ be upward closed set systems in the lattice of $\{0,1\}^n$. The celebrated Harris-Kleitman inequality implies that if $|X|=\alpha 2^n$, $|Y|=\beta 2^n$, the density of the set of points in exactly one of $X$ and $Y$ is maximal when $X$ and $Y$ are independent, meaning $|X\cap Y|=\alpha\beta 2^n$. Is the same true of three upward closed systems, $X$, $Y$, and $Z$? Suppose $|X|=|Y|=|Z|$. Kahn asked whether the set of points in exactly one of $X$, $Y$, $Z$ has density at most $\frac49$. We answer this question in the negative.