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Statement

Setting (p. 1). For an integer $k\ge3$, a regular $(k-1)$-simplex is a set of $k$ points that are pairwise equidistant. $S_d^k(n)$ is the largest number of regular $(k-1)$-simplices spanned by $n$ points of $\mathbb R^d$, and $T_d(n)$ is the largest number of equilateral triangles determined by $n$ points of $\mathbb R^d$, so that $T_d(n)=S_d^3(n)$ (p. 2). Triangles and simplices of every side length are counted together.

Theorem 2 (p. 2). Let $d$ and $k$ be fixed integers with $d\ge 2k\ge 6$, and put $r=\lfloor d/2\rfloor$. Then

$$S_d^k(n)=\binom rk\Bigl(\frac nr\Bigr)^k+o(n^k).$$

Conjecture 1 (p. 1), which the paper attributes to Erdős's 1994 paper in Math. Pannon. (its reference [12]), quoted: "Any $n$ points in $\mathbb{R}^6$ can span at most $n^3/27+o(n^3)$ equilateral triangles, i.e., $T_6(n)\leq n^3/27+o(n^3)$." The paper notes (p. 2) that Theorem 2 with $d=6$ and $k=3$ gives it, since $r=3$ and $\binom33(n/3)^3=n^3/27$. The paper also recalls (p. 1) the Erdős--Purdy lower bound $T_6(n)\ge n^3/27-O(n^2)$ from $n$ points spread evenly over three pairwise orthogonal circles, so in fact $T_6(n)=n^3/27+o(n^3)$.

For even $d$, Corollary 6 sharpens the error term to $\Theta(n^{k-1})$.

Proof pointer

§ 4, p. 9. The key step is Lemma 17 (p. 9): for integers $d\ge2k\ge3$ and $r=\lfloor d/2\rfloor$, the $k$-uniform hypergraph whose vertices are points of $\mathbb R^d$ and whose edges are the regular $(k-1)$-simplices among them contains no copy of $H_{r+1}^{(k)}(3)$, the $3$-blowup of the hypergraph $H_{r+1}^{(k)}$ obtained from $K_{r+1}$ by adding $k-2$ new vertices to each edge (Definition 8, p. 6). A copy would give, by Lemma 15 (p. 8), $r+1$ pairwise orthogonal affine spaces of dimension at least $2$ inside $\mathbb R^d$, which is impossible since $d<2(r+1)$. Mubayi's theorem $\mathrm{ex}(n,H_{r+1}^{(k)})=\binom rk(n/r)^k+o(n^k)$ (Lemma 9, p. 6) and the standard fact that blowing up changes the Turán number by $o(n^k)$ give the upper bound (display (3), p. 6). The lower bound is the construction of § 2: $r$ pairwise orthogonal unit circles with a common center and $n$ points spread over them as evenly as possible, every two points on different circles being at distance $\sqrt2$. The proof on p. 9 opens "Let $r=\lfloor d/2\rfloor\geq k\geq 6$ be fixed integers" [sic]; the theorem's hypothesis is $d\ge2k\ge6$.

Read depth

Claims checked: the definitions, Conjecture 1, Theorem 2, Lemma 17 and its proof (p. 9) were read clause by clause on the page images of the print (arXiv version 4). Lemmas 9 and 15 were read as statements only. Nothing here is independently reviewed.

Dependencies

Lemma 9 is Mubayi's theorem (the paper's [18]); the blowup estimate is cited as well known (the paper's [17]); Lemma 15 is proved in the paper (p. 8) from Lemma 14 (p. 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 $d=6$, $k=3$ bounds the number of equilateral triangles of all side lengths together spanned by $n$ points of $\mathbb R^6$ by $n^3/27+o(n^3)$; the problem asks for this bound for triangles of side $1$ only, which are among those counted, so the theorem gives the problem's bound.