Problem #654: Let $x_1,\ldots,x_n\in \mathbb{R}^2$ with no four points on...

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Progress

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 $x_j$.