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 $n\times n$ 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.