A Correlation Inequality on Three Functions

2025/02/20 by Williams, Kada
#Combinatorics (math.CO) #FOS: Mathematics

paper · doi:10.48550/arxiv.2502.14857

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|=α2ⁿ, |Y|=β2ⁿ, the density of the set of points in exactly one of X and Y is maximal when X and Y are independent, meaning |X∩ Y|=αβ2ⁿ. 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.

Related

• A general correlation inequality for level sets of sums of independent random variables using the Bernoulli part with applications to the almost sure local limit theorem
• Pair Correlations in Uniform Countable Sets
• Second-order bounds on correlations between increasing families
• Quantitative Correlation Inequalities via Semigroup Interpolation
• On the Correlation of Increasing Families