On the number of regular simplices in odd dimensions
Abstract
Let Skd(n) denote the maximum number of regular (k−1)-simplices spanned by n points in ℝd. For all fixed r≥ k≥3, the exact value of Sk2r(n) was recently determined by Dumitrescu and the authors for all sufficiently large n when k=3, and conditionally on an optimization problem when k≥4. In this paper we establish a sharp estimate in odd dimensions by proving that Sk2r+1(n)=binomrk(n/r)k+Θ(nk−2/3) for all fixed r≥ k≥3. This can be viewed as a generalization of the result by Erdős and Pach on unit distances in odd dimensions. We also establish a structural characterization of nearly extremal point sets. A notable difference from the even-dimensional case is that nearly extremal point sets in odd dimensions admit two distinct configurations. The proof leverages techniques from hypergraph Turán theory and linear algebra, with additional geometric arguments.
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BibTeX
@misc{clemen2026number,
title = {{On the number of regular simplices in odd dimensions}},
author = {Felix Christian Clemen and Dingyuan Liu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.26801},
eprint = {2609.26801},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.26801},
}
RIS
TY - UNPB
TI - On the number of regular simplices in odd dimensions
AU - Felix Christian Clemen
AU - Dingyuan Liu
PY - 2026
DA - 2026/09/07
PB - arXiv
DO - 10.48550/arxiv.2609.26801
UR - https://arxiv.org/abs/2609.26801
AB - Let S^k_d(n) denote the maximum number of regular (k−1)-simplices spanned by n points in ℝ^d. For all fixed r≥ k≥3, the exact value of S^k_2r(n) was recently determined by Dumitrescu and the authors for all sufficiently large n when k=3, and conditionally on an optimization problem when k≥4. In this paper we establish a sharp estimate in odd dimensions by proving that S^k_2r+1(n)=binomrk(n/r)^k+Θ(n^k−2/3) for all fixed r≥ k≥3. This can be viewed as a generalization of the result by Erdős and Pach on unit distances in odd dimensions. We also establish a structural characterization of nearly extremal point sets. A notable difference from the even-dimensional case is that nearly extremal point sets in odd dimensions admit two distinct configurations. The proof leverages techniques from hypergraph Turán theory and linear algebra, with additional geometric arguments.
ER -