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 This result generalizes an unpublished result of Berge from circa 1980.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 F⊂2[n] is intersection-union (IU) if for any A,B∈F we have 1≤|A∩B|≤n−1. Using the aforementioned result, we derive several exact product- and sum-type results for IU-families.