Let $h(n)$ be such that any $n$ points in $\mathbb{R}^2$, with no three on a line and no four on a circle, determine at least $h(n)$ distinct distances. Does $h(n)/n\to \infty$?

Erdős could not even prove $h(n)\geq n$. Pach has shown $h(n) < n^{\log_2 3}$. Erdős, Füredi, and Pach [EFPR93] have improved this to $h(n) < n\exp(c\sqrt{\log n})$ for some constant $c>0$.