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 This result generalizes an unpublished result of Berge from circa 1980.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.
翻译:在本文中, 我们显示, 根据两套下设置( 简易综合体), 它们之间有一个匹配匹配的不连接集, 覆盖两套下设置中较小部分。 这个结果概括了1980年circa 的Berge未公布的结果。 结果对交叉分割的家庭和 Chv\'atal 的猜想具有很好的轮廓。 更具体地说, 我们显示, Chv\'atal 的猜想对覆盖2美元的交叉家庭来说是真实的。 一个家庭 $\mathcal F\subset 2\\\\ { n} $ 是交叉组合( IU), 如果对于任何 A, B\\ in\ mathcal F$, 我们有$ 1\le {A\ cap B ⁇ le n-1$。 我们使用上述结果为IU- families 得出了几种精确的产品和总类结果 。