Problem 654

Erdős Problem #654: declared status 'open'. Formalized: no. 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 $x_i$? Current best: In [Er87b] and [ErPa90] Erd\H{o}s 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 $x_j$. Prize: no. Tags: distances, geometry.