Problem #132: Let $A\subset \mathbb{R}^2$ be a set of $n$ points

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Progress

Asked by Erdős and Pach. Hopf and Pannowitz [HoPa34] proved that the largest distance between points of $A$ can occur at most $n$ times, but it is unknown whether a second such distance must occur.

It may be true that there are at least $n^{1-o(1)}$ many such distances. In [Er97e] Erdős offers \$100 for 'any nontrivial result'.

Erdős [Er84c] believed that for $n\geq 5$ there must always exist at least two such distances. This is false for $n=4$, as witnessed by two equilateral triangles of the same side-length glued together. Erdős and Fishburn [ErFi95] proved this is true for $n=5$ and $n=6$.

Clemen, Dumitrescu, and Liu [CDL25] have proved that there always at least two such distances if $A$ is in convex position (that is, no point lies inside the convex hull of the others). They also prove it is true if the set $A$ is 'not too convex', in a specific technical sense.

See also [223], [756], and [957].