Rainbow matchings in properly-colored hypergraphs

TL;DR

This paper proves the existence of large rainbow matchings in properly-colored hypergraphs under certain size and coloring conditions, extending previous results related to the Erdős Matching Conjecture.

Contribution

It establishes new conditions guaranteeing rainbow matchings in properly-colored hypergraphs, generalizing prior work on the Erdős Matching Conjecture.

Findings

Existence of rainbow matchings under specified conditions

Generalization of Erdős Matching Conjecture results

Conditions on hypergraph size and coloring for matchings

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 $n\geq3k^{2}s$, and $e(H_{i})>{{n}\choose {k}}-{{n-s+1}\choose {k}}$, then there exists a rainbow matching of size $s$, containing one edge from each $H_i$. This generalizes some previous results on the Erd\H{o}s Matching Conjecture.