Rainbow matchings for 3-uniform hypergraphs

Abstract

Kühn, Osthus, and Treglown and, independently, Khan proved that if $H$ is a $3$-uniform hypergraph with $n$ vertices such that $n\in 3\mathbb{Z}$ and large, and ${\delta }_1(H)>\left(\genfrac{}{}{0ex}{}{n-1}{2}\right)-\left(\genfrac{}{}{0ex}{}{2n/3}{2}\right)$, then $H$ contains a perfect matching. In this paper, we show that for $n\in 3\mathbb{Z}$ sufficiently large, if $F_1,\dots ,F_{n/3}$ are 3-uniform hypergrapghs with a common vertex set and ${\delta }_1(F_i)>\left(\genfrac{}{}{0ex}{}{n-1}{2}\right)-\left(\genfrac{}{}{0ex}{}{2n/3}{2}\right)$ for $i\in [n/3]$, then $\{F_1,\dots ,F_{n/3}\}$ admits a rainbow matching, i.e., a matching consisting of one edge from each $F_i$. This is done by converting the rainbow matching problem to a perfect matching problem in a special class of uniform hypergraphs.