We prove that any family $E_1, \ldots , E_{\lceil rn \rceil}$ of (not necessarily distinct) sets of edges in an $r$-uniform hypergraph, each having a fractional matching of size $n$, has a rainbow fractional matching of size $n$ (that is, a set of edges from distinct $E_i$'s which supports such a fractional matching). When the hypergraph is $r$-partite and $n$ is an integer, the number of sets needed goes down from $rn$ to $rn-r+1$. The problem solved here is a fractional version of the corresponding problem about rainbow matchings, which was solved by Drisko and by Aharoni and Berger in the case of bipartite graphs, but is open for general graphs as well as for $r$-partite hypergraphs with $r>2$. Our topological proof is based on a result of Kalai and Meshulam about a simplicial complex and a matroid on the same vertex-set.

中文翻译：

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彩虹分数匹配

我们证明了在 $r$-uniform 超图中的任何族 $E_1, \ldots , E_{\lceil rn \rceil}$（不一定是不同的）边集，每个边都有一个大小为 $n$ 的分数匹配，具有大小为 $n$ 的彩虹分数匹配（即支持这种分数匹配的不同 $E_i$ 的一组边）。当超图为 $r$-partite 且 $n$ 为整数时，所需的集合数从 $rn$ 减少到 $rn-r+1$。这里解决的问题是关于彩虹匹配的相应问题的分数版本，在二部图的情况下由 Drisko 和 Aharoni 和 Berger 解决，但对一般图以及 $r$-partite hypergraphs 是开放的$r>2$。我们的拓扑证明基于 Kalai 和 Meshulam 关于同一个顶点集上的单纯复形和拟阵的结果。