  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
\cite{drisko} (as generalized by the first two authors \cite{ab}), 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 \cite{CamWan}, that every Latin square has a diagonal
(permutation submatrix) 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.
