Problem 91

Erdős Problem #91: declared status 'open'. Formalized: yes. 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. Current best: In [Er87b] Erd\H{o}s says that there are at least two non-similar examples for $6\leq n\leq 9$. Prize: no. OEIS: A186704. Tags: distances, geometry.