Minimum $\ell$-degree thresholds for rainbow perfect matching in $k$-uniform hypergraphs

Abstract

Given $n\in k\mathbb{N}$ elements set $V$ and $k$-uniform hypergraphs ${\mathcal{H}}_1,\dots ,{\mathcal{H}}_{n/k}$ on $V$. A rainbow perfect matching is a collection of pairwise disjoint edges $E_1\in {\mathcal{H}}_1,\dots ,E_{n/k}\in {\mathcal{H}}_{n/k}$ such that $E_1\cup \cdots \cup E_{n/k}=V$. In this paper, we determine the minimum $\ell$-degree condition that guarantees the existence of a rainbow perfect matching for sufficiently large $n$ and $\ell \ge k/2$.