Rainbow matchings in properly-colored hypergraphs

Abstract

A hypergraph $H$ is properly colored if for every vertex $v\in V(H)$, all the edges incident to $v$ have distinct colors. In this paper, we show that if $H_1$, \cdots, $H_s$ are properly-colored $k$-uniform hypergraphs on $n$ vertices, where $n\ge 3k^{2}s$, and $e(H_i)>\left(\genfrac{}{}{0ex}{}{n}{k}\right)-\left(\genfrac{}{}{0ex}{}{n-s+1}{k}\right)$, then there exists a rainbow matching of size $s$, containing one edge from each $H_i$. This generalizes some previous results on the Erdős Matching Conjecture.