AI 中文总结
研究完全\(r\)部\(r\)均匀超图中\(k\)匹配的\(t\)相交族问题，用两种方法确定其最大规模并刻画极值族，一种借助排列的\(t\)相交族结果得定理，另一种基于\(t\)覆盖法进行补充刻画。
AI 中文摘要
给定整数\(1\leq t\leq k\)，在完全\(r\)部\(r\)均匀超图中，一族\(k\)匹配若任意两个成员至少共享\(t\)条公共边，则称其为\(t\)相交的。此概念统一了几类经过充分研究的相交族。本文采用两种方法确定\(k\)匹配的\(t\)相交族的最大规模并刻画达到此界的极值族。利用凯勒等人关于排列的\(t\)相交族的最新结果，得到阈值仅依赖于\(t\)的埃尔德什 - 柯 - 拉多型定理，还开发了基于\(t\)覆盖的方法对极值族进行补充刻画。
英文摘要
Given integers $1\leq t\leq k$, a family of $k$-matchings in a complete $r$-partite $r$-uniform hypergraph is said to be $t$-intersecting if any two of its members share at least $t$ common edges. This concept unifies several well-studied classes of intersecting families, including classical intersecting families, intersecting families of permutations, partial permutations, and generalized permutations, as well as intersecting families of injections. In this paper we employ two approaches to determine the maximum size of $t$-intersecting families of $k$-matchings and to characterize the extremal families that attain this bound. Using a recent result of Keller, Lifshitz, Minzer, and Sheinfeld on $t$-intersecting families of permutations, we obtain Erdős-Ko-Rado-type theorems whose thresholds depend only on $t$. We also develop a $t$-cover-based approach that offers a complementary characterization of the extremal families.
