Let A be a set of $2m$ points on the plane so that no open set of diameter $2$ has more than m of them. Define $A+A+...+A$ ($k$ times) to be the multiset of $k$-sums from $A$. That is, we consider all possible $(2m)^k$ sums.
Question. Does there exists a number $k$ independent of $m$ such that the points in the latter multiset can be paired so that the distance between the points each pair is at least 2?
Conjecture. $k=2$ is enough.
Comment. $k=1$ is not enough for the following reason. Take three points at distance $2$ and a point in the middle of them - then there is no such pairing. The intuition here is that the points get further and further apart when we are summing and hence we shall have the desired pairing.
One remark: The claim is true if $k$ is large enough for the following reason - the number of points in any open bounded open set divided by the total number of points goes to zero (concentration function bounds from probability) and so for $k$ large enough one can make any set of diameter 4 to have at most half of the points. But now notice that the graph on the multiset of the sums formed by adding an edge between two sums iff the distance between them is at least 2 has degree at least half the number of the sums as otherwise we would have more than half of the points would fit in a set of diameter 4 (the neighbourhood of that point with low degree). But this is Dirac's condition as so we have a perfect matching and are done.