On the rainbow matching conjecture for 3-uniform hypergraphs

Abstract

Aharoni and Howard and, independently, Huang et al. (2012) proposed the following rainbow version of the Erdős matching conjecture: For positive integers n, k and m with n ⩾ km, if each of the families \(F_{1},\ldots,F_{m}\subseteq\left(\begin{array}{c}[n]\\ k\end{array}\right)\) has size more than \(\max\{\left(\begin{array}{c}n\\ k\end{array}\right)-\left(\begin{array}{c}n-m+1\\ k\end{array}\right),\left(\begin{array}{c}km-1\\ k\end{array}\right)\}\), then there exist pairwise disjoint subsets e1,…,em such that ei ∈ Fi for all i ∈ [m]. We prove that there exists an absolute constant n0 such that this rainbow version holds for k = 3 and n ⩾ n0. 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. (2012) to find an almost regular subgraph of H − V(M); 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 Łuczak and Mieczkowska (2014) and might be of independent interest.