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. Estimate $g_d(n)$ . In particular, does

\lim_{d\to \infty}\frac{g_d(n)}{d^{n-1}}

exist?

Source: erdosproblems.com/1089

Geometry
Distances

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SolvedProved

An accepted solution exists. The statement is true.

SOLVED, in the site's label: for $n\ge2$ , $\binom{d+1}{n-1}+1\le g_d(n)\le\binom{d+n-1}{n-1}+1$ , so the limit exists and equals $1/(n-1)!$ . The site credits the lower bound to the Aletheia agent of Feng et al. [Fe26] and the upper bound to Bannai, Bannai and Stanton [BBS83]; the limit question is thereby answered yes, and the accepted proof is recorded on its claim page (Feng, 2026). The derived standing, solved and proved, is more specific than the site's label, which names no polarity: the accepted claim proves that the limit exists. The exact value of $g_d(n)$ is not determined.

BBS83: Bannai, Eiichi and Bannai, Etsuko and Stanton, Dennis, An upper bound for the cardinality of an $s$ -distance subset in real Euclidean space. II. Combinatorica 3 (1983), 147-152.
Cr62: Croft, H. T., $9$ -point and $7$ -point configurations in $3$ -space. Proc. London Math. Soc. (3) 12 (1962), 400-424.
BB81: Bannai, Eiichi and Bannai, Etsuko, An upper bound for the cardinality of an $s$ -distance subset in real Euclidean space. Combinatorica 1 (1981), no. 2, 99-102, as [Fe26] cites it. Not on the site's list; Remark 4.4 of [Fe26] reports that its Remark 3(ii) already answers the problem.
Er75f: Erdős, Paul, On some problems of elementary and combinatorial geometry. Ann. Mat. Pura Appl. (4) 103 (1975), 99-108.
Fe26: T. Feng et al, Semi-Autonomous Mathematics Discovery with Gemini: A Case Study on the Erdős Problems. arXiv:2601.22401 (2026).

Problem 1089

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