Year: 2026
Type: preprint
Abstract: Let $S^k_d(n)$ denote the maximum number of regular $(k-1)$-simplices spanned by $n$ points in $\mathbb{R}^d$. For all fixed $r\geq k\geq3$, 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 $k\geq4$. In this paper we establish a sharp estimate in odd dimensions by proving that \[S^k_{2r+1}(n)=\binom{r}{k}\left(\frac{n}{r}\right)^k+Θ(n^{k-2/3})\] for all fixed $r\geq k\geq3$. 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.
Source: arXiv (Cornell University)
Authors: Clemen, Felix Christian, Liu, Dingyuan
Language: English
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7214164020