On the rainbow matching conjecture for 3-uniform hypergraphs

Aharoni and Howard and, independently, Huang et al. (2012) proposed the following rainbow version of Erd?s matching conjecture: For positive integers n, k m with n ? km, if each 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 exists an absolute constant n0 this holds = 3 n0. convert problem to a on special hypergraph H. combine several existing techniques matchings in uniform hypergraphs: Find absorbing M H; use randomization process Alon find almost regular subgraph H ? V(M); perfect V(M). To complete process, we also need new result 3-uniform hypergraphs, which can be viewed as stability ?uczak Mieczkowska (2014) might independent interest.