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We are given arbitrary 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 $T\in S$.

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 $T\in S$.

Given two sets $A$ and $B$, members of $A$ have been split into subsets Si which maybe they have intersections with each other and union of them maybe not equal to A determine whether there is a mapping of members of $A$ to $B$ so that each member of  $B$ has been mapped by at most $k$ members of $A$ that are in one subset Si.

We are given arbitrary 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 $T\in S$.