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Claim. For every $d\ge4$ there is $n_0(d)$ such that for every $n\ge n_0(d)$, every set of $n$ points of $\mathbb R^d$ with diameter one in which the distance one occurs $f_d(n)$ times is a Lenz configuration of a specified type: its points lie on $\lfloor d/2\rfloor$ circles in mutually orthogonal planes, with one circle replaced by a sphere when $d$ is odd, whose radii $r_i$ satisfy $r_i^2+r_j^2=1$. The exact value of $f_d(n)$ in Problem 223 follows for all $d\ge4$ and $n\ge n_0(d)$, together with the extremal configurations.

Covers. The exact value of $f_d(n)$ and the extremal sets for every $d\ge4$ and every $n$ sufficiently large in terms of $d$. The paper also treats the maximum number of unit distances in the same regime. Nothing is claimed about $d=2$, $d=3$, or small $n$.

The argument. The proof is a stability analysis of the Lenz configuration combined with extremal graph theory, refining the asymptotic $(\frac{p-1}{2p}+o(1))n^2$ that Erdős obtained from the absence of a complete $(p+1)$-partite unit-distance graph with parts of size three. The library's [[../library/distance_problems/swanepoel_2009_unit_distances_diameters_euclidean_spaces/_index|card for the paper]] states the two structure theorems and the corollary on exact values.

Acceptance. The paper is refereed: K. J. Swanepoel, Unit distances and diameters in Euclidean spaces, Discrete & Computational Geometry 41 (2009), no. 1, 1–27, published online 2008-05-08; the preprint is arXiv:0707.0213, posted 2007-07-02. The curator of erdosproblems.com, Thomas Bloom, marks the problem solved and credits this paper with the exact description of $f_d(n)$ for $d\ge4$ and large $n$.