How Many Colors Guarantee a Rainbow Matching?

Given a coloring of the edges of a multi-hypergraph, a rainbow t-matching is a collection of t disjoint edges, each having a different color. In this note we study the problem of finding a rainbow t-matching in an r-partite r-uniform multi-hypergraph whose edges are colored with f colors such that every color class is a matching of size t. This problem was posed by Aharoni and Berger, who asked to determine the minimum number of colors which guarantees a rainbow matching. We improve on the known upper bounds for this problem for all values of the parameters. In particular for every fixed r, we give an upper bound which is polynomial in t, ∗The first author was supported by DFG within the research training group Methods for Discrete Structures. †The second author was supported in part by SNSF grant 200021-149111 and by a USA-Israel BSF grant. ‡The third author was partially supported by DFG within the research training group Methods for Discrete Structures. the electronic journal of combinatorics 21(1) (2014), #P1.27 1 improving the superexponential estimate of Alon. Our proof also works in the setting not requiring the hypergraph to be r-partite.