Multicolored matchings in hypergraphs

For a collection of (not necessarily distinct) matchings M = (M₁, M₂,:::, M_q) in a hypergraph, where each matching is of size t, a matching M of size t contained in the union [^qi=1^Mi is called a rainbow matching if there is an injective mapping from M to M assigning to each edge e of M a matching M_i 2 M containing e. Let f(r, t) denote the maximal 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) 2^r–1 (t – 1), proved that the equality holds for r = 2 as well as for t = 2 and conjectured that the 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 into some of those results.