Erdős Problem 652

### What is the conjecture

https://www.erdosproblems.com/652

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 $k\to \infty$?

Status: solved

### Choose either option
- [ ] I plan on working on this conjecture
- [x] This issue is up for grabs: I would like to see this conjecture added by somebody else