OPEN
This is open, and cannot be resolved with a finite computation.
Let $x_1,\ldots,x_n\in \mathbb{R}^2$ with no four points on a circle. Must there exist some $x_i$ with at least $(1-o(1))n$ distinct distances to other $x_i$?
It is clear that every point has at least $\frac{n-1}{3}$ distinct distances to other points in the set.
In
[Er87b] and
[ErPa90] Erdős and Pach ask this under the additional assumption that there are no three points on a line (so that the points are in general position), although they only ask the weaker question whether there is a lower bound of the shape $(\tfrac{1}{3}+c)n$ for some constant $c>0$.
They suggest the lower bound $(1-o(1))n$ is true under the assumption that any circle around a point $x_i$ contains at most $2$ other $x_j$.
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This page was last edited 02 October 2025.
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T. F. Bloom, Erdős Problem #654, https://www.erdosproblems.com/654, accessed 2026-01-16
