Perfect matchings in down-sets

01/11/2022

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by   Peter Frankl, et al.

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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 This result generalizes an unpublished result of Berge from circa 1980.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.