Erdős Problem #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 \(k\to \infty\)? Mathialagan [Ma21] proved that given a set \(P\) of \(k\) points and a set \(Q\) of \(n\) points, with \(2\leq k\leq n^{1/3}\), there exists a point in \(P\) which determines \(\gg (kn)^{1/2}\) distances to points in \(Q\).