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

$$\lim_{n\to\infty}\frac{f_d(n)}{n^2}=\frac12-\frac1{2p}=\frac{p-1}{2p},$$

so $f_d(n)=\bigl(\frac{p-1}{2p}+o(1)\bigr)n^2$ in the notation of Problem 223. The same limit holds for the maximum number of times any single distance can occur among $n$ points of $\mathbb R^d$.

Covers. The order of growth and the leading constant of $f_d(n)$ for every $d\ge4$. The exact value of $f_d(n)$ for finite $n$ is not claimed; Swanepoel's later result, on its own claim page in this folder, determines it for all large $n$.

The argument. The lower bound places about $n/p$ points on a quarter arc (the points with nonnegative coordinates) of each of $p$ mutually orthogonal circles of radius $1/\sqrt2$, generalizing Lenz's construction: every two points on different circles are at distance one, and the restriction to a quarter arc keeps two points on the same circle within distance one, so the set has diameter one. 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, $K_{p+1}(3)$; its $p+1$ triangles lie in $p+1$ mutually orthogonal planes, which need $2p+2>d$ dimensions. The library's [[../library/distance_problems/erdos_1960_sets_distances_points_euclidean_space/_index|card for the paper]] records the main theorem, the Lenz construction and the three-dimensional bounds the paper also proves.

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. The curator of erdosproblems.com, Thomas Bloom, marks the problem solved and credits the case $d\ge4$ to this paper.