Problem Statement
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.
Categories:
Geometry Distances
Progress
For $n=3$ the equilateral triangle is the only such set. For $n=4$ the square or two equilateral triangles sharing an edge give two non-similar examples.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'. (In [Er87b] this is described as 'a colleague from Zagreb (unfortunately I do not have his letter)'.) A published proof of this fact is provided by Kovács [Ko24c].
In [Er87b] Erdős says that there are at least two non-similar examples for $6\leq n\leq 9$.
The minimal possible number of distinct distances is the subject of [89].
Source: erdosproblems.com/91 | Last verified: January 13, 2026