A Correlation Inequality on Three Functions
Kada Williams
TL;DR
This work investigates whether the Harris-Kleitman two-set correlation inequality extends to three upward closed families $X$, $Y$, and $Z$ on the Boolean lattice when they have equal densities. It derives bounds on the density of the set $S_1$ of points contained in exactly one of the three families, using Harris-Kleitman inequalities, a multi-slice decomposition, and measure-preserving mappings on weighted cubes. The key results include a concrete counterexample showing that the equal-density conjecture fails (even in small dimensions) and an upper bound of about $0.515$ for $|S_1|/2^n$, along with a larger counterexample to Kahn's conjecture in a high-dimensional weighted cube, highlighting a gap between known bounds and extremal configurations. The work also discusses a weighted-poset framework and opens questions about finer triple-density thresholds in the Boolean lattice.
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 $|X|=α2^n$, $|Y|=β2^n$, the density of the set of points in exactly one of $X$ and $Y$ is maximal when $X$ and $Y$ are independent, meaning $|X\cap Y|=αβ2^n$. 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.