Suppose we have an $(n,n)$-bipartite graph with edges colored with $k$ colors. Is anything known about the existence of rainbow matchings (i.e. a matching that uses each color exactly once, for $k=n$) for a random bipartite graph (e.g. that for $k$ colors and more than $f(k,n)$ edges we get a rainbow matching with $p \rightarrow 1$)?

In the noncolored case, Hall's theorem makes proving this kind of results relatively simple, since we are interested in the non-existence of "no matching possible" witness (i.e. a subset that violates Hall condition) and we can use union bound to bound the probability from above (for $A_k$ = "k-th subset is a witness" give bound to $\mathbb{P}(\cup A_k)$). However, there is no simple condition of this kind equivalent to the existence of a rainbow matching.