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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 $\mathbb R^d$ in which the unit distance 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 two-sphere when $d$ is odd, whose radii $r_i$ satisfy $r_i^2+r_j^2=1$. As a corollary, the exact value of $f_d(n)$ in Problem 1085 is determined for every even $d\ge6$ and every $n\ge n_0(d)$.

Covers. The exact value of $f_d(n)$, with the extremal configurations, for every even $d\ge6$ and every $n$ sufficiently large in terms of $d$. For odd $d\ge5$ the structure theorem holds but the exact value is not determined: it depends on the maximum number of unit distances among $n$ points of a two-sphere, which is open. Nothing is claimed about $d=2$, $d=3$, $d=4$ (Brass's exact value, a pending claim on its own claim page in this folder) or small $n$.

Depends on. No page of this wiki.

The argument. The proof is a stability analysis of the Lenz configuration combined with extremal graph theory, refining the asymptotic $(\frac{p-1}{2p}+o(1))n^2$ of Erdős's 1960 paper and the additive-constant estimate of his 1967 paper, both on their own claim pages in this folder. 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; the same paper's diameter theorem is the accepted claim for the higher dimensions of Problem 223.

Acceptance. 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. Not reviewed: the site's remarks say that this paper determined $f_d(n)$ exactly for even $d\ge6$, but the site labels the problem OPEN, so the remark is not an acceptance of the problem or of a part.