Co-degree threshold for rainbow perfect matchings in uniform hypergraphs

TL;DR

This paper determines the minimum degree condition needed for rainbow perfect matchings in large uniform hypergraphs, extending known results on perfect matchings and introducing new techniques.

Contribution

It establishes the degree threshold for rainbow perfect matchings in uniform hypergraphs and develops a novel absorbing method for the proof.

Findings

Identified the exact degree threshold for rainbow perfect matchings.

Extended existing results on perfect matchings to rainbow settings.

Developed a new absorbing technique for hypergraph matchings.

Abstract

Let $k$ and $n$ be two integers, with $k\geq 3$, $n\equiv 0\pmod k$, and $n$ sufficiently large. We determine the $(k-1)$-degree threshold for the existence of a rainbow perfect matchings in $n$-vertex $k$-uniform hypergraph. This implies the result of R\"odl, Ruci\'nski, and Szemer\'edi on the $(k-1)$-degree threshold for the existence of perfect matchings in $n$-vertex $k$-uniform hypergraphs. In our proof, we identify the extremal configurations of closeness, and consider whether or not the hypergraph is close to the extremal configuration. In addition, we also develop a novel absorbing device and generalize the absorbing lemma of R\"odl, Ruci\'nski, and Szemer\'edi.