Degree Conditions for Matchability in 3-Partite Hypergraphs

We study conjectures relating degree conditions in 3-partite hypergraphs to the matching number of the hypergraph, and use topological methods to prove special cases. In particular, we prove a strong version of a theorem of Drisko [14] (as generalized by the first two authors [2]), that every family of 2n-1 matchings of size n in a bipartite graph has a partial rainbow matching of size n. We show that milder restrictions on the sizes of the matchings suffice. Another result that is strengthened is a theorem of Cameron and Wanless [11], that every n×n Latin square has a generalized diagonal (set of n entries, each in a different row and column) in which no symbol appears more than twice. We show that the same is true under the weaker condition that the square is row-Latin.