Erdős problem 655

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?

The answer is no: as Zach Hunter observed, the regular n-gon (n points equally spaced on a circle) is valid and determines only ⌊n/2⌋ < (1+c)n/2 distinct distances, for every c > 0. (In the spirit of related conjectures of Erdős and others, presumably some kind of assumption that the points are in general position was intended; see erdos_655.variants.general_position.)

The disproof — the regular n-gon construction together with its supporting lemmas — is formalised at the linked commit.