A Correlation Inequality on Three Functions

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 $|X|=\hat{I}\pm 2^n$, $|Y|={\hat{I}}^2{2}^n$, the density of the set of points in exactly one of $X$ and $Y$ is maximal when $X$ and $Y$ are independent, meaning $|X\cap Y|=\hat{I}\pm {\hat{I}}^2{2}^n$. 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 $\frac{4}{9}$. We answer this question in the negative.