Erdős Problem #1082 — Distinct Distances from Points in General Position

(i) Let $A\subset\mathbb R^2$ be a finite set with no three points on a line. Must $A$ determine at least $\lfloor |A|/2\rfloor$ distinct distances?

(ii) No. The following assertion is false: for every nonempty finite set $A\subset\mathbb R^2$ with no three points on a line, there exists a point $a\in A$ such that the number of distinct distances from $a$ to the other points of $A$, denoted $\operatorname{distinctDistancesFrom}(A,a)-1$, is at least $\lfloor |A|/2\rfloor$; that is,

$$\lfloor |A|/2\rfloor\leq \operatorname{distinctDistancesFrom}(A,a)-1.$$