Some results 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, studying it with techniques from convex optimization.