A Correlation Inequality on Three Functions

Abstract: Let $X$ and $Y$ be upward closed set systems in the lattice of ${0,1}<sup>n$. The celebrated Harris-Kleitman inequality implies that if $|X|=\alpha 2<sup>n$, $|Y|=\beta 2<sup>n$, the density of the set of points in exactly one of $X$ and $Y$ is maximal when $X$ and $Y$ are independent, meaning $|X\cap Y|=\alpha\beta 2<sup>n$. 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.