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Statement

Setting (p. 16). $T_d^{\mathrm{unit}}(n)$ is the largest number of unit equilateral triangles spanned by $n$ points of $\mathbb R^d$; $\mathbb 1_P$ is $1$ when $P$ holds and $0$ otherwise.

Proposition 25 (pp. 16--17). Let $r\ge3$ be a fixed integer. For every sufficiently large $n$,

$$T_{2r}^{\mathrm{unit}}(n)=\sum_{1\le i<j<k\le r}n_in_jn_k +\sum_{i\in[r]}(n_i-\mathbb 1_{n_i\notin4\mathbb Z})(n-n_i),$$

with $(n_1,\ldots,n_r)$ the splitting of $n$ chosen in Theorem 3.

The expression is that of Theorem 3 without the terms $\frac{n_i-p_i}3+\mathbb 1_{p_i>8}(p_i-8)$, which count the triangles with all three vertices on one circle, whose side differs from the others'. For $r=3$ it is $n^3/27+O(n^2)$, since the parts differ from $n/3$ by at most $2$.

Proof pointer

P. 17, a sketch only: the paper says the upper bound follows by an argument analogous to that of Theorems 3 and 5 with the triangles on one circle left uncounted, and the lower bound from the Lenz construction of § 2.2 without those triangles. In that construction, two points on different circles of radius $1$ are at distance $\sqrt2$, so the counted triangles have side $\sqrt2$; scaling the circles to radius $1/\sqrt2$ makes them unit triangles with the same count. No detailed proof is printed.

Read depth

Claims checked: the definition and Proposition 25 were read clause by clause on the page images of pp. 16--17 (arXiv version 4), with the sketch that follows. The paper prints no full proof, so none was checked. Nothing here is independently reviewed.

Dependencies

Theorem 3, Theorem 5 and, through them, 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

• Problem 755: the case $r=3$ is the problem's count, triangles of side $1$ among $n$ points of $\mathbb R^6$, and gives $n^3/27+O(n^2)$ for it for large $n$; the paper states it with a proof sketch only. The problem's bound also follows from Theorem 2, which counts triangles of all sizes.