Problem Statement
For which $n$ are there $n$ points in $\mathbb{R}^2$, no three on a line and no four on a circle, which determine $n-1$ distinct distances and so that (in some ordering of the distances) the $i$th distance occurs $i$ times?
Categories:
Geometry Distances
Progress
An example with $n=4$ is an isosceles triangle with the point in the centre. Erdős originally believed this was impossible for $n\geq 5$, but Pomerance constructed a set with $n=5$ (see [Er83c] for a description), and Palásti has proved such sets exist for all $n\leq 8$.Erdős believed this is impossible for all sufficiently large $n$. This would follow from $h(n)\geq n$ for sufficiently large $n$, where $h(n)$ is as in [98].
Source: erdosproblems.com/217 | Last verified: January 14, 2026