论文标题
完美的搭配
Perfect matchings in down-sets
论文作者
论文摘要
在本文中，我们表明，给定两个下集（简单的复合物），它们之间有一个匹配的匹配，可以匹配不相交的集合，并覆盖了两个下组中的较小。该结果概括了大约1980年的伯格（Berge）的未发表的结果。结果具有很好的推论，可以使家庭和奇瓦塔尔（Chvátal）的猜想进行。更具体地说，我们表明，Chvátal的猜想对于与$ 2 $的覆盖家庭相交的家庭是正确的。 一个家庭$ \ MATHCAL F \ subset 2^{[n]} $是相交 - Union（IU），如果对于任何$ a，b \ in \ Mathcal f $，我们有$ 1 \ le | a \ cap b | \ cap b | \ le n-1 $。使用上述结果，我们得出了IU家族的几个精确的产物和汇总结果。
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 This result generalizes an unpublished result of Berge from circa 1980.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.
