Perfect matchings in down-sets

TL;DR

This paper proves a new matching theorem for down-sets (simplicial complexes) that extends Berge's 1980 result, with implications for intersecting families and Chvátal's conjecture, including confirming it for families with covering number 2.

Contribution

It generalizes Berge's matching result to down-sets, providing new insights and proofs related to Chvátal's conjecture and intersecting family structures.

Findings

Established a matching between two down-sets covering the smaller one.

Confirmed Chvátal's conjecture for intersecting families with covering number 2.

Derived exact product- and sum-type results for intersection-union families.

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 This result generalizes an unpublished result of Berge from circa 1980.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 Chv\'atal's conjecture. More concretely, we show that Chv\'atal'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.