On the rainbow matching conjecture for 3-uniform hypergraphs

TL;DR

This paper proves a rainbow matching conjecture for 3-uniform hypergraphs when the number of vertices exceeds a certain constant, using advanced hypergraph matching techniques and a new stability result.

Contribution

The paper establishes the rainbow matching conjecture for 3-uniform hypergraphs for sufficiently large n, introducing new methods and a stability result for hypergraph matchings.

Findings

Proved the rainbow matching conjecture for 3-uniform hypergraphs with large enough n.

Developed a new stability result for matchings in 3-uniform hypergraphs.

Combined existing hypergraph matching techniques with new results to achieve the proof.

Abstract

Aharoni and Howard, and, independently, Huang, Loh, and Sudakov proposed the following rainbow version of Erd\H{o}s matching conjecture: For positive integers $n,k,m$ with $n\ge km$, if each of the families $F_1,\ldots, F_m\subseteq {[n]\choose k}$ has size more than $\max\{\binom{n}{k} - \binom{n-m+1}{k}, \binom{km-1}{k}\}$, then there exist pairwise disjoint subsets $e_1,\dots, e_m$ such that $e_i\in F_i$ for all $i\in [m]$. We prove that there exists an absolute constant $n_0$ such that this rainbow version holds for $k=3$ and $n\geq n_0$. We convert this rainbow matching problem to a matching problem on a special hypergraph $H$. We then combine several existing techniques on matchings in uniform hypergraphs: find an absorbing matching $M$ in $H$; use a randomization process of Alon et al. to find an almost regular subgraph of $H-V(M)$; and find an almost perfect matching in $H-V(M)$. To complete the process, we also need to prove a new result on matchings in 3-uniform hypergraphs, which can be viewed as a stability version of a result of {\L}uczak and Mieczkowska and might be of independent interest.