Let $A\subset \mathbb{R^2}$ be a finite set such that $|A|=k^2$. Let $x_i\in \mathbb{R^2}$, $i=1,2,3,4$, be four points in the plane in general position (no three lie on any line).

Let us form the multiset of cardinality $4k^2$ out of the four translates $A+x_i$ and call it $M$.

Question: Can we partition $M$ into four sets (not multisets) $A_i$, $i=1,2,3,4$ with cardinalities $(k+1)^2$, $(k-1)(k+3)$, $(k-1)^2$, $(k-1)^2$, respectively?

Question: Are there any good references for problems of this kind?