On Kleitman's Conjecture

Abstract

Chvátal conjectured that amongst the largest intersecting subfamilies of a finite subset-closed family of sets is a star. Kleitman later strengthened Chvátal's conjecture, defining a partial ordering on the vector space freely generated by $2^{[n]}$ and suggesting that the vector of every maximal intersecting subfamily of $2^{[n]}$ is bigger than a convex combination of stars. We restate Kleitman's conjecture in terms of the cochain complex of the discrete cube, describing it as the optimization of a convex objective. We examine the pseudoinverse of the codifferential, along with its relation to monotonicity, through which we recover the Harris-Kleitman inequality and give a generalization of a theorem of Frankl and Kupavskii on perfect matchings in superset-closed families. We show level families to satisfy Kleitman's conjecture, providing explicit cochains. We further show maximal intersecting families in the union of two stars and nonnegative threshold families to satisfy Kleitman's conjecture. We close with several strengthenings of the conjecture and related open problems.