 
Summary: Multicolored matchings in hypergraphs
Noga Alon
Abstract
For a collection of (not necessarily distinct) matchings M = (M1, M2, . . . , Mq) in a hypergraph,
where each matching is of size t, a matching M of size t contained in the union t
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 rpartite runiform hypergraph, such that there is no rainbow matching of size t.
Aharoni and Berger showed that f(r, t) 2r1
(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.
1 Introduction
A matching in a hypergraph is a collection of pairwise disjoint edges. For a collection of (not necessarily
distinct) matchings M = (M1, M2, . . . , Mq) in a hypergraph, where each matching is of size t, a
matching M of size t contained in the union t
