Problem Statement
Let $f(n)$ be minimal such that every set of $n$ points in $\mathbb{R}^2$ contains at most $f(n)$ many sets of four points which are 'degenerate' in the sense that some pair are the same distance apart. Estimate $f(n)$ - in particular, is it true that $f(n)\leq n^{3+o(1)}$?
Categories:
Geometry Distances
Progress
A question of Erdős and Purdy [ErPu71], who proved\[n^3\log n \ll f(n) \ll n^{7/2}.\]Source: erdosproblems.com/1087 | Last verified: January 19, 2026