Multicolored matchings in hypergraphs

For a collection of (not necessarily distinct) matchingsM = (M1,M2, . . . ,Mq) in a hypergraph, where each matching is of size t, a matching M of size t contained in the union ∪i=1Mi is called a rainbow matching if there is an injective mapping from M to M assigning to each edge e of M a matching Mi ∈M containing e. Let f(r, t) denote the maximum k for which there exists a collection of k matchings, each of size t, in some r-partite r-uniform hypergraph, such that there is no rainbow matching of size t. Aharoni and Berger showed that f(r, t) ≥ 2r−1(t − 1), proved that equality holds for r = 2 as well as for t = 2 and conjectured that equality holds for all r, t. We show that in fact f(r, t) is much bigger for most values of r and t, establish an upper bound and point out a relation between the problem of estimating f(r, t) and several results in additive number theory, which provides new insights on some such results.