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Source. Theorem 4, p. 4, of Konrad J. Swanepoel, Unit distances and diameters in Euclidean spaces, Discrete Comput. Geom. 41 (2009), no. 1, 1--27, doi:10.1007/s00454-008-9082-x; labels and pages are those of arXiv:0707.0213v1 (2 July 2007), the version named on the source card; the proof is on pp. 18-19.

Read depth. Claims checked: the statement and the Stability Theorem it uses (p. 18) were read clause by clause on the printed pages, and the proof was followed. Nothing here is independently reviewed.

Statement

Theorem 4 (p. 4). Let $d\ge4$ be even and $p=d/2$. For each $\varepsilon>0$ there are $\delta>0$ and $N$ such that every set of $n\ge N$ points in $\mathbb R^d$ with at least $(\frac{p-1}{2p}-\delta)n^2$ unit distance pairs can be partitioned into $S_0,S_1,\ldots,S_p$ with $|S_0|<\varepsilon n$ and, for each $i=1,\ldots,p$,

$$\frac np-\varepsilon n<|S_i|<\frac np+\varepsilon n,$$

where each $S_i$ lies on a circle $C_i$ and the circles $C_1,\ldots,C_p$ have a common centre and are mutually orthogonal.

The theorem does not fix the radii; that the radii of mutually unit-distant circles satisfy $r_i^2+r_j^2=1$ is Lemma 8 (p. 9).

Proof pointer

Pp. 18-19. By Lemma 8 (p. 9), the unit distance graph contains no complete $(p+1)$-partite graph with three vertices in each class, so the Erdős-Simonovits Stability Theorem, as stated on p. 18, partitions the set into $S_0,\ldots,S_p$ of the right sizes with each point of $S_i$ joined to all but fewer than $\varepsilon n$ points outside $S_i$. If some $S_i$ had four non-concyclic points, these with three points from each other class would span a complete $p$-partite unit distance graph forcing them, by Lemma 8, onto a circle; so each $S_i$ is concyclic, and Lemma 8 again makes the circles concentric and mutually orthogonal.

Dependencies

The Erdős-Simonovits stability theorem (cited from Bollobás, Extremal Graph Theory, Chapter 5, Theorem 4.2); Lemma 8 (p. 9), whose proof the paper omits as easy.

Bears on

• Problem 1085 and Problem 223: an input to Theorem 1 for even $d$; on its own it describes near-extremal sets and fixes no value of either problem's $f_d(n)$.