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Claim. For every $d\ge4$, with $p=\lfloor d/2\rfloor$,

$$\lim_{n\to\infty}\frac{f_d(n)}{n^2}=\frac{p-1}{2p},$$

so $f_d(n)=\bigl(\frac{p-1}{2p}+o(1)\bigr)n^2$ in the notation of Problem 1085. The paper proves the same limit for the maximum number of times the diameter occurs among $n$ points of diameter one, which is Problem 223.

Covers. The order of growth and the leading constant of $f_d(n)$ for every $d\ge4$, even or odd. The lower-order terms are not claimed: Erdős's 1967 paper, Brass's and Swanepoel's exact values for even $d$ and Erdős and Pach's second-order term for odd $d\ge5$, each on its own claim page in this folder, refine this estimate. Nothing is claimed for $d=2$ or $d=3$, where the paper proves $c_1n^{4/3}<f_3(n)<c_2n^{5/3}$ and records the planar bounds then known.

Depends on. No page of this wiki.

The argument. The lower bound is Lenz's construction: $n/p$ points on each of $p$ mutually orthogonal circles of radius $1/\sqrt2$, so that every two points on different circles are at distance one, give $\binom p2(n/p)^2=\frac{p-1}{2p}n^2+O(n)$ unit distances. The upper bound applies the Erdős–Stone theorem to the graph of unit-distance pairs: more than $(\frac{p-1}{2p}+\varepsilon)n^2$ edges would force a complete $(p+1)$-partite subgraph with parts of size three, whose $p+1$ triangles lie in mutually orthogonal planes and need more than $d$ dimensions. The library's [[../library/distance_problems/erdos_1960_sets_distances_points_euclidean_space/_index|card for the paper]] records the main theorem, the construction and the three-dimensional bounds.

Acceptance. The paper is a journal publication: P. Erdős, On sets of distances of $n$ points in Euclidean space, Magyar Tud. Akad. Mat. Kutató Int. Közl. 5 (1960), 165–169. Not reviewed: the site's remarks credit the upper bound to this paper and the lower bound to Lenz, but the site labels the problem OPEN, so the remark is not an acceptance of the problem or of a part.