Problem 655

Erdős Problem #655: declared status 'open'. Formalized: yes. Let $x_1,\ldots,x_n\in \mathbb{R}^2$ be such that no circle whose centre is one of the $x_i$ contains three other points. Are there at least$$(1+c)\frac{n}{2}$$distinct distances determined between the $x_i$, for some constant $c>0$ and all $n$ sufficiently large? Prize: no. Tags: distances, geometry.