Global and local degree conditions for matchability

Abstract

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''.