Erdős Problem #654 — Let $x_1,\ldots,x_n\in \mathbb{R}^2$ with no four points on a circle. Must there exist some $x_i$ with at least $(1-o(1))n$ distinct distances to other $x_i$?

Erdős asked whether some point must determine nearly all distances, and Erdős–Pach separately asked whether one can force a bound exceeding one third of the points under general-position assumptions. The problem remains open in that stronger geometric setting.

Known results

• The universal trivial bound is $f(n)\geq (n-1)/3$.
• Erdős (1997) proposed the nearly linear bound, while calling it possibly too optimistic.
• Erdős–Pach (1987, 1990) formulated the general-position version seeking $f(n)>(1/3+c)n$ for fixed $c>0$.

Aletheia construction, 2026

The Erdős Problems record credits Aletheia with a construction of $n$ points having no four concyclic, while every point determines at most $3n/4$ distinct distances. Since the points lie on two lines, this refutes the nearly-$n$ conjecture but neither settles the general-position version nor disproves an improved bound $f(n)>(1/3+c)n$.

Current status (as of March 2026): The nearly-linear conjecture is reported false, but the general-position question and the possibility of some fixed improvement over $(n-1)/3$ remain open.