Problem 652

Erdős Problem #652: declared status 'proved'. Formalized: no. 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 $\alpha_k$ be minimal such that, for all large enough $n$, there exists a set of $n$ points with $R(x_k)<\alpha_kn^{1/2}$. Is it true that $\alpha_k\to \infty$ as $k\to \infty$? Prize: no. Tags: distances, geometry.