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Statement

Setting. $e_r(S)$ is as in Theorem A, and Lenz configurations and their associated partitions are as defined on p. 5 and restated on the Theorem B page.

Theorem C (p. 6). Let $d\ge4$ and $p=\lfloor d/2\rfloor$. For every $\varepsilon>0$ there are $\delta>0$ and $n_0\in\mathbb{N}$ such that, for every $S\subset\mathbb{R}^d$ with $|S|=n\ge n_0$ and every $r\colon S\to(0,\infty)$ with

$$e_r(S)>\Bigl(1-\frac1p-\delta\Bigr)n^2,$$

there are $T\subseteq S$ and $c>0$ with $|T|<\varepsilon n$, $S\setminus T$ a Lenz configuration with distance $c$, and $r$ identically $c$ on $S\setminus T$. Moreover the associated partition $S_1,\ldots,S_p$ of $S\setminus T$ has $\frac np-\varepsilon n<|S_i|<\frac np+\varepsilon n$ for every $i\in[p]$.

It is the favourite-distance analogue of Theorem 4 (p. 6, cited to Swanepoel's paper on unit distance and diameter graphs), the same statement for $u(S)>\frac12\bigl(1-\frac1p-\delta\bigr)n^2$ with no function $r$.

Proof pointer

Section 5, pp. 6--12. The constants are fixed with $\delta<\varepsilon^2/144$ and small enough for Theorem 4 at the level $32\delta$. The double-edge decomposition of the proof of Theorem A, combined with Theorem 4, gives the result quickly for $d\ge6$. Dimensions $4$ and $5$ take most of the section: the paper derives $d=4$ from $d=5$ by viewing $\mathbb{R}^4$ as a hyperplane of $\mathbb{R}^5$ and treats $d=5$ by a longer counting argument, which it attributes (p. 6) to complications in the extremal theory of digraphs rather than to the Lenz construction.

Read depth

Claims checked: Theorem C was read clause by clause on the page image of p. 6 of the arXiv preprint; the proof in Section 5 was read for structure only. Theorem 4 is cited, not proved, in the paper. Nothing here is independently reviewed.

Dependencies

None in the corpus. External inputs named by the paper: Theorem 4, the stability theorem for unit distances; Theorems 1 and 2; the Erdős--Stone theorem; bounds for $f_3(n)$; and lemmas of Avis, Erdős and Pach excluding orientations of a complete $4$-partite graph from favourite distance digraphs in $\mathbb{R}^5$.

Source. K. J. Swanepoel, Favorite distances in high dimensions, in Thirty Essays on Geometric Graph Theory (J. Pach, ed.), Algorithms and Combinatorics 29, Springer, New York, 2013, 499--519; read in the arXiv preprint arXiv:1108.4817 (24 August 2011), whose labels and pages are used here; see the source card.

Bears on

None directly. The theorem describes near-extremal configurations; Problem 754's bound comes from Theorem A.