Erdős Problem #1089 — Let $g_d(n)$ be minimal such that every collection of $g_d(n)$ points in $\mathbb{R}^d$ determines at least $n$ many distinct distances.

Kelly posed the question, recorded by Erdős in 1975. It asks for the growth of $g_d(n)$ and the existence of its normalized limit.

Known results

• For $n\ge 2$, $\binom{d+1}{n-1}+1\le g_d(n)\le\binom{d+n-1}{n-1}+1$ (Bannai, Bannai, and Stanton, 1983; lower bound attributed to Aletheia, [Fe26]).
• Hence $\displaystyle\lim_{d\to\infty}g_d(n)/d^{n-1}=1/(n-1)!$ for $n\ge 2$; also $g_d(1)=2$.
• Special cases include $g_1(3)=4$, $g_2(3)=6$, and $g_3(3)=7$ (Croft, 1962).

Independent rediscovery by Aletheia

Aletheia independently derived the asymptotic result; human experts then identified an earlier solution in Bannai–Bannai, 1981, Remark 3(ii). The result is corroborated by the cited bounds and published discussion.

Current status (as of February 2026): The limit is settled for $n\ge 2$, with value $1/(n-1)!$; $n=1$ is also settled, while finer questions such as exact values remain separate.