Erdős Problem #604

declared status 'open'. Formalized: no. 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 $\gg n/\sqrt{\log n}$? Current best: The best known bound is$$\gg n^{c-o(1)},$$due to Katz and Tardos [KaTa04], where$$c=\frac{48-14e}{55-16e}=0.864137\cdots.$$ References [Er75f] Erd\H{o}s, Paul, On some problems of elementary and combinatorial geometry. Prize: $500. Tags: distances, geometry.

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