We are given finite sets $A$ and $B$ and a set $S\subseteq \mathcal{P}(A)$. The members of $\mathcal{S}$ may have arbitrary intersections with one another and their union is not necessarily $A$.  We wish to determine whether there is a function $A\to B$  so that no member of $B$ is the image of more than $k$ members of any $T\in S$.