匹配性的全局与局部度条件

中文摘要 AI 辅助

霍尔婚姻定理的一个推论是，对于集合列表 $(V_1, \ldots,V_m)$ 存在不同代表系的充分条件为：对每个 $i\in [m]$ 和 $v \in \bigcup_{i\in [m]}V_i$，有 $|V_i|\ge deg_{\{V_1, \ldots,V_m\}}(v)$。我们称此为全局条件。一个民间结果是，局部条件——即不等式仅对满足 $v \in V_i$ 的配对 $i,v$ 成立——就足够了。这些是一般类型结果的特殊情况：大的集合，其元素在某种意义上是稀疏的，具有一个在相关图中独立的代表系。我们研究两种此类场景，在两种场景中，每个 $V_i$ 被一个 $k$-均匀超图 $H_i$ 替换，代表为超边，不同性被不相交性取代。在一种设置中，稀疏性通过超图中顶点的度来衡量；在另一种设置中，通过线图中顶点的度来衡量。证明使用拓扑版本的霍尔定理。特别地，我们将使用由向量表示定义的图的独立复形的拓扑连通性的下界。我们还提供了局部无限版本的简短证明，即著名的“米尔纳-谢拉赫定理”。

英文摘要

A corollary of Hall's marriage theorem is that a sufficient condition for a list $(V_1, \ldots ,V_m)$ of sets to have a system of distinct representatives is that $|V_i|\ge deg_{\{V_1, \ldots ,V_m\}}(v)$ for every $i\in [m]$ and $v \in \bigcup_{i\in [m]}V_i$. This we dub a {\em global} condition. A folklore result is that a {\em local} condition - that the inequality holds for pairs $i,v$ for which $v \in V_i$ - suffices. These are special cases of a general type of results - large sets, whose elements are sparse in some sense, have a system of representatives that is independent in a related graph. We study two such scenarios, in both of which each $V_i$ is replaced by a $k$-uniform hypergraph $H_i$, the representatives are hyperedges, and distinctness is replaced by disjointness. In one setting the sparsity is measured by the degrees of vertices in the hypergraphs, in the other by the degrees of vertices in the line graph. The proofs use the topological version of Hall's theorem. In particular, we shall use a lower bound on the topological connectivity of the independence complex of a graph, defined by vector representations. We also provide short proofs of the local infinite version, known as the ``Milner-Shelah theorem''.

发表机构

• Technion（以色列理工学院）
• Haifa University（海法大学）

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