Perfect matchings in down-sets

Abstract

In this paper, we show that, given two down-sets (simplicial complexes) there is a matching between them that matches disjoint sets and covers the smaller of the two down-sets. This result generalizes an unpublished result of Berge from circa 1980. The result has nice corollaries for cross-intersecting families and Chvátal's conjecture. More concretely, we show that Chvátal's conjecture is true for intersecting families with covering number $2$. A family $\mathcal{F}\subset 2^{[n]}$ is intersection-union (IU) if for any $A,B\in \mathcal{F}$ we have $1\le |A\cap B|\le n-1$. Using the aforementioned result, we derive several exact product- and sum-type results for IU-families.