Let k>1, and let F be a family of 2n+kβ3 non-empty sets of
edges in a bipartite graph. If the union of every k members of F
contains a matching of size n, then there exists an F-rainbow
matching of size n. Upon replacing 2n+kβ3 by 2n+kβ2, the result can be
proved both topologically and by a relatively simple combinatorial argument.
The main effort is in gaining the last 1, which makes the result sharp