Problem 653

Erdős Problem #653: declared status 'open'. Formalized: yes. Let $x_1,\ldots,x_n\in \mathbb{R}^2$ and let $R(x_i)=\#\{ \lvert x_j-x_i\rvert : j\neq i\}$, where the points are ordered such that$$R(x_1)\leq \cdots \leq R(x_n).$$Let $g(n)$ be the maximum number of distinct values the $R(x_i)$ can take. Is it true that $g(n) \geq (1-o(1))n$? Prize: no. Tags: distances, geometry.