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Multicolored matchings in hypergraphs For a collection of (not necessarily distinct) matchings M = (M1, M2, . . . , Mq) in a hypergraph,
 

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 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.
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

  

Source: Alon, Noga - School of Mathematical Sciences, Tel Aviv University

 

Collections: Mathematics