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Source. Theorem 5, 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 p. 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 5 (p. 4). Let $d\ge5$ be odd and $p=\lfloor d/2\rfloor$. For each $\varepsilon>0$ there are $\delta>0$ and $N$ such that every set $S$ 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 $S_1$ lies on a $2$-sphere $\Sigma_1$, each $S_i$, $i=2,\ldots,p$, lies on a circle $C_i$, and $\Sigma_1,C_2,\ldots,C_p$ have a common centre and are mutually orthogonal.

Proof pointer

P. 19. As for Theorem 4, the Stability Theorem (p. 18), applied with $\varepsilon/5$, gives the partition, and Lemma 8 (p. 9) puts each $S_i$ on a $2$-sphere. If two classes were not concyclic, four non-concyclic points from each with three from every other class would span at least $3+3+2(p-2)=d+1$ dimensions in mutually orthogonal subspaces, a contradiction; after moving fewer than $4\varepsilon n/5$ points into $S_0$, Lemma 8 makes the sphere and 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 odd $d$; on its own it describes near-extremal sets and fixes no value of either problem's $f_d(n)$.