Erdős #670 · geometry / distances

The lower bound of $\binom{n}{2}$ for the diameter is trivial. Erdős [Er97f] proved the claim when $d=1$. The quantifiers are slightly ambiguous, but likely Erdős had in mind the regime where $d$ is fixed and the $o(1)$ term $\to 0$ as $n\to \infty$ (perhaps at a rate which depends on $d$).

This was disproved if $n$ grows with $d$ by Ho [Ho26]: there exist infinitely many $n$ and, with $d=n^2-n$, a set $A\subset \mathbb{R}^d$ such that all pairwise distances differ by at least $1$, with diameter at most

The source labels this entry OPEN in the 2026-09-08 snapshot. Editorial progress notes are reproduced as source commentary, not independently verified results. Prize amounts and Lean badges at the source are not local awards or verification.