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Problem 604 — Erdős Problems

Problem 604

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: erdosproblems.com/604

Geometry
Distances

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Er75f: Erdős, Paul, On some problems of elementary and combinatorial geometry. Ann. Mat. Pura Appl. (4) (1975), 99-108.
Er97e: Erdős, Paul, Some of my favourite unsolved problems. Math. Japon. (1997), 527-537.
KaTa04: Katz, Nets Hawk and Tardos, Gábor, A new entropy inequality for the Erdős distance problem. Towards a theory of geometric graphs (2004), 119-126.

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