Problem Statement
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 $h(n)/n\to \infty$?
Categories:
Geometry Distances
Progress
Erdős could not even prove $h(n)\geq n$. Pach has shown $h(n)<n^{\log_23}$. Erdős, Füredi, and Pach [EFPR93] have improved this to\[h(n) < n\exp(c\sqrt{\log n})\]for some constant $c>0$.Source: erdosproblems.com/98 | Last verified: January 13, 2026