Let [n]r 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]r contains a matching of size k. We conjecture the following rainbow version of this observation: if F 1,F 2,. . .,F k ⊆ [n]r are of size larger than (k−1)n r−1 then there exists a rainbow matching, that is, a choice of disjoint edges f i ∈ F i . We prove this conjecture for r=2 and r=3.

Paul Erdős has conjectured that Menger's theorem extends to infinite graphs in the following way: whenever A, B are two sets of vertices in an infinite graph, there exist a set of disjoint A−B paths and an A−B separator in this graph such that the separator consists of a choice of precisely one vertex from each of the paths. We prove this conjecture for graphs that contain a set of disjoint paths to B from all but countably many vertices of A. In particular, the conjecture is true when A is countable.

Conjecture: Let [Mscr] and [Nscr] be two matroids (possibly of infinite ranks) on the same set S. Then there exists a set I independent in both [Mscr] and [Nscr], which can be partitioned as I=H∪K, where sp[Mscr](H)∪sp[Nscr](K)=S. This conjecture is an extension of Edmonds' matroid intersection theorem to the infinite case. We prove the conjecture when one of the matroids (say [Mscr]) is the sum of countably many matroids of finite rank (the other matroid being general). For the proof we have also to answer the following question: when does there exist a subset of S which is spanning for [Mscr] and independent in [Nscr]?