Size Conditions for the Existence of Rainbow Matchings

Let f(n, r, k) be the minimal number such that every hypergraph larger than f(n, r, k) contained in ([n] r ) contains a matching of size k, and let g(n, r, k) be the minimal number such that every hypergraph larger than g(n, r, k) contained in the r-partite r-graph [n]r contains a matching of size k. The Erdős-Ko-Rado theorem states that f(n, r, 2) = (n−1 r−1 ) (r ≤ n 2 ) and it is easy to show that g(n, r, k) = (k − 1)nr−1. The conjecture inspiring this paper is that if F1, F2, . . . , Fk ⊆ ([n] r ) are of size larger than f(n, r, k) or F1, F2, . . . , Fk ⊆ [n]r are of size larger than g(n, r, k) then there exists a rainbow matching, i.e. a choice of disjoint edges fi ∈ Fi. In this paper we deal mainly with the second part of the conjecture, and prove it for the cases r ≤ 3 and k = 2. The proof of the r = 3 case uses a Hall-type theorem on rainbow matchings in bipartite graphs. For the proof of the k = 2 case we prove a Kruskal-Katona type theorem for r-partite hypergraphs. We also prove that for every r and k there exists n0 = n0(r, k) such that the r-partite version of the conjecture is true for n > n0.