Clemen 2025 number regular simplices higher dimensions

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corollary_6: Clemen, Dumitrescu and Liu's second-order term in even dimensions: for fixed r >= k >= 3, the maximum number of regular (k-1)-simplices spanned by n points of R^{2r} is binom(r,k)(n/r)^k + Theta(n^{k-1}).

proposition_25: Clemen, Dumitrescu and Liu's count for one side length: for fixed r >= 3 and all sufficiently large n, the maximum number of unit equilateral triangles spanned by n points of R^{2r} equals the Theorem 3 expression without its term for triangles lying on one circle; the paper sketches the proof.

theorem_2: Clemen, Dumitrescu and Liu's asymptotic theorem: for fixed integers d >= 2k >= 6 and r = floor(d/2), the maximum number of regular (k-1)-simplices spanned by n points of R^d is binom(r,k)(n/r)^k + o(n^k); the case d = 6, k = 3 gives Erdős's conjecture T_6(n) <= n^3/27 + o(n^3).

theorem_3: Clemen, Dumitrescu and Liu's exact count: for fixed r >= 3 and all sufficiently large n, the maximum number of equilateral triangles spanned by n points of R^{2r} is an explicit cubic expression in a near-balanced partition of n into r parts; when 12r divides n it equals binom(r,3)(n/r)^3 + (r-1)n^2/r + n/3.

theorem_5: Clemen, Dumitrescu and Liu's reduction for k >= 4: for fixed r >= k >= 4 and all sufficiently large n, the maximum number of regular (k-1)-simplices spanned by n points of R^{2r} equals the maximum of an explicit polynomial count f_k over the splittings of n into r parts.

theorem_7: Clemen, Dumitrescu and Liu's stability theorem: for fixed r >= k >= 3, a set of n points of R^{2r} spanning S_{2r}^k(n) - o(n^k) regular (k-1)-simplices has all but o(n) of its points on r pairwise orthogonal circles with a common center and a common radius, n/r - o(n) on each.

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Felix Christian Clemen, Adrian Dumitrescu, Dingyuan Liu, The number of regular simplices in higher dimensions. arXiv:2507.19841 (2025); the edition read is version 4 (28 July 2026), whose labels and pages are cited below.

The paper studies S_d^k(n), the maximum number of regular (k-1)-simplices spanned by n points in R^d. Theorem 2 shows that for fixed d >= 2k >= 6, writing r = floor(d/2), S_d^k(n) = binom(r,k) (n/r)^k + o(n^k); for k = 3 and d = 6 this gives T_6(n) = n^3/27 + o(n^3) and so proves Erdos's conjecture (Conjecture 1, p. 1) that n points in R^6 span at most n^3/27 + o(n^3) equilateral triangles. Theorem 3 goes further and gives the exact value of T_{2r}(n) for every even d = 2r >= 6 and all sufficiently large n, via an explicit near-balanced partition (n_1,...,n_r) of n; Corollary 4 specializes it to n divisible by 12r, Theorem 5 reduces S_{2r}^k(n) for k >= 4 to maximizing an explicit polynomial, and Corollary 6 gives S_{2r}^k(n) = binom(r,k)(n/r)^k + Theta(n^{k-1}) for r >= k >= 3. The main tool for these exact results is the stability result Theorem 7: an almost-extremal set in R^{2r} lies, up to o(n) points, on r pairwise orthogonal circles with common center and radius, about n/r points on each, the shape of the Erdos-Purdy construction with points spread evenly over three pairwise orthogonal circles (which gave T_6(n) >= n^3/27 - O(n^2)). The proof leverages hypergraph Turan theory together with linear algebra. In its concluding remarks the paper states Proposition 25, the exact maximum number of unit equilateral triangles in R^{2r} for large n, with a proof sketch only. Problem 755 asks for the bound n^3/27 + o(n^3) only for equilateral triangles of side 1 in R^6; T_6(n) counts triangles of every size, so the case d = 6, k = 3 of Theorem 2 gives the problem's bound.

Source: https://arxiv.org/abs/2507.19841. The arXiv record names arXiv's non-exclusive distribution license (arXiv:2507.19841), every other right reserved.

Read status: claims checked for Conjecture 1, Theorems 2, 3, 5 and 7, Corollaries 4 and 6 and Proposition 25, read clause by clause on the page images of version 4; the proofs of Theorem 2 (with Lemma 17) and Corollary 6 followed, the proofs of Theorems 3, 5 and 7 read for structure only, and Proposition 25 has only a printed sketch. Nothing here is independently reviewed.

Bears on. #755: Theorem 2 (p. 2) with d = 6, k = 3 bounds the number of equilateral triangles of all sizes together among n points of R^6 by n^3/27 + o(n^3), which contains the problem's bound for side 1; Theorem 3 gives that count exactly for large n, and Proposition 25 (pp. 16--17, proof sketched) gives the exact count for side 1 alone.

Results.

• Theorem 2 (p. 2), with Conjecture 1 (p. 1) and Lemma 17 (p. 9): S_d^k(n) = binom(r,k)(n/r)^k + o(n^k) for fixed d >= 2k >= 6, r = floor(d/2).
• Theorem 3 and Corollary 4 (p. 2): the exact value of T_{2r}(n) for fixed r >= 3 and large n.
• Theorem 5 (p. 2): S_{2r}^k(n) = max f_k(n_1,...,n_r) for fixed r >= k >= 4 and large n.
• Corollary 6 (p. 3): S_{2r}^k(n) = binom(r,k)(n/r)^k + Theta(n^{k-1}) for fixed r >= k >= 3.
• Theorem 7 (p. 3): stability of almost-extremal sets in R^{2r}.
• Proposition 25 (pp. 16--17): the exact maximum number of unit equilateral triangles in R^{2r}, r >= 3, n large.

No file of this source is held: no license on record permits its redistribution, and the card cites the edition it names above.