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Statement

Setting (pp. 1--3). $S_{2r}^k(n)$ is the largest number of regular $(k-1)$-simplices (sets of $k$ pairwise equidistant points) spanned by $n$ points of $\mathbb R^{2r}$. Two circles are orthogonal when the affine planes they span are orthogonal, that is, when the associated linear subspaces are orthogonal (pp. 2--3).

Theorem 7 (p. 3). Let $r\ge k\ge3$ be fixed integers, and let $X\subseteq\mathbb R^{2r}$ be a set of $n$ points spanning $S_{2r}^k(n)-o(n^k)$ regular $(k-1)$-simplices. Then $X$ splits into disjoint parts $A_0,A_1,\ldots,A_r$ with $|A_0|=o(n)$ and $|A_i|=n/r-o(n)$ for each $i\in[r]$, and there are pairwise orthogonal circles $C_1,\ldots,C_r\subseteq\mathbb R^{2r}$ such that

• (1) $A_i\subseteq C_i$ for every $i\in[r]$, and
• (2) the circles $C_1,\ldots,C_r$ have one common center and one common radius.

The paper calls it the main tool for its exact results (p. 3).

Proof pointer

§ 5, pp. 10--11. By Theorem 2 the set spans $\binom rk(n/r)^k-o(n^k)$ regular simplices, and by Lemma 17 its simplex hypergraph has no copy of $H_{r+1}^{(k)}(3)$; the stability Lemma 10 (p. 6, deduced from Pikhurko's stability theorem and the hypergraph removal lemma) then gives an $r$-partite subhypergraph with $\binom rk(n/r)^k-o(n^k)$ edges, whose parts have $n/r-o(n)$ vertices by Lemma 13. Claim 18 (p. 10) finds in each part a set $A_i$ of $n/r-o(n)$ points spanning a plane and lying on a circle, using Lemma 15 and the dimension count $2r$; Claim 19 (p. 11) shows the circles are pairwise orthogonal, using Erdős's bound $\mathrm{ex}(n,K^{(k)}_{3,\ldots,3})=o(n^k)$, and concentric of equal radius, using Lemma 16.

Read depth

Claims checked: the definitions and Theorem 7 were read clause by clause on the page image of p. 3 (arXiv version 4). The proof (pp. 10--11) was read for structure only. Nothing here is independently reviewed.

Dependencies

Theorem 2 with its Lemma 17; Lemmas 10, 13, 15 and 16 of the paper (pp. 6--8); Pikhurko's stability theorem (Lemma 11, the paper's [20]), the hypergraph removal lemma of Rödl, Nagle, Skokan, Schacht and Kohayakawa (Lemma 12, the paper's [23]) and Erdős's theorem on complete $k$-partite hypergraphs (the paper's [10]).

Source. F. C. Clemen, A. Dumitrescu and D. Liu, The number of regular simplices in higher dimensions, arXiv:2507.19841 (2025), read in version 4 (28 July 2026); see the source card.

Bears on

• Problem 755: with $r=k=3$ it describes the near-extremal sets of points of $\mathbb R^6$ for equilateral triangles of all sizes together as, up to $o(n)$ points, the Erdős--Purdy configuration of three pairwise orthogonal concentric circles of equal radius carrying about $n/3$ points each; it is the tool behind the exact counts, not itself a bound.