---

Statement

Setting (pp. 1--2). $S_{2r}^k(n)$ is the largest number of regular $(k-1)$-simplices (sets of $k$ pairwise equidistant points) spanned by $n$ points of $\mathbb R^{2r}$. The paper takes every variable of such a function to be a nonnegative integer (p. 2). $\binom{S}{m}$ is the family of $m$-element subsets of $S$, $[r]=\{1,\ldots,r\}$, and $\mathbb 1_P$ is $1$ when $P$ holds and $0$ otherwise.

Theorem 5 (p. 2). Let $r\ge k\ge4$ be fixed integers and let $n$ be sufficiently large. Then

$$S_{2r}^k(n)=\max_{n_1+\cdots+n_r=n}f_k(n_1,\ldots,n_r),$$

where

$$f_k(n_1,\ldots,n_r)=\sum_{\mathcal I\in\binom{[r]}{k}}\prod_{i\in\mathcal I}n_i +\sum_{1\le\ell\le\lfloor k/2\rfloor}\ \sum_{\mathcal J\in\binom{[r]}{\ell}} \Bigl(\prod_{j\in\mathcal J}(n_j-\mathbb 1_{n_j\notin4\mathbb Z})\Bigr) \Bigl(\sum_{\mathcal I\in\binom{[r]\setminus\mathcal J}{k-2\ell}}\prod_{i\in\mathcal I}n_i\Bigr).$$

The first sum counts simplices with at most one vertex on each of $r$ orthogonal circles, the second those using $\ell$ pairs at distance $\sqrt2$ on $\ell$ different circles (§ 2.2, pp. 4--5). The paper remarks (p. 2) that the maximum is attained with $|n_i-n/r|=O(1)$ for every $i$, and does not identify the maximizer, which it says appears to depend intricately on $r$ and $k$.

Proof pointer

§ 6.1, pp. 12--14. The lower bound (1) (p. 5) is the even-dimensional Lenz construction of § 2.2. For the upper bound, Theorem 7 places all but $o(n)$ points of an extremal set on $r$ pairwise orthogonal concentric circles of equal radius; Claim 20 (p. 12) removes the exceptional points by an exchange argument, and Claim 21 (p. 13) bounds the count by $\max f_k$, giving display (6) (p. 14) for every $r\ge k\ge3$, where for $k=3$ the function carries a further term for triangles on one circle. Theorem 5 is the case $k\ge4$, in which no regular simplex has three vertices on one circle.

Read depth

Claims checked: Theorem 5 and the definition of $f_k$ were read clause by clause on the page images of pp. 2 and 4 (arXiv version 4). The proof was read for structure only. Nothing here is independently reviewed.

Dependencies

Theorem 7.

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

None. The theorem concerns regular simplices with $k\ge4$ vertices; the triangle case behind Problem 755 is Theorem 3.