be the complete r-partite hypergraph with vertex classes of size n. It is an easy exercise to show that every set of more than (k−1)n
r−1 edges in [n]
contains a matching of size k. We conjecture the following rainbow version of this observation: if F
2,. . .,F
are of size larger than (k−1)n
r−1 then there exists a rainbow matching, that is, a choice of disjoint edges f
. We prove this conjecture for r=2 and r=3.