OPEN
Suppose $A\subset \mathbb{R}^2$ has $\lvert A\rvert=n$ and minimises the number of distinct distances between points in $A$. Prove that for large $n$ there are at least two (and probably many) such $A$ which are non-similar.
For $n=5$ the regular pentagon is the unique such set (which has two distinct distances). Erdős mysteriously remarks in
[Er90] this was proved by 'a colleague'. A published proof of this fact is provided by Kovács
[Ko24c].
