OPEN

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?

A problem of Erdős and Pach. It is easy to see that this assumption implies that there are at least $\frac{n-1}{2}$ distinct distances determined by every point.