The number of regular simplices in higher dimensions

TL;DR

This paper investigates the maximum number of regular simplices formed by points in high-dimensional space, providing asymptotic bounds and exact counts for specific cases, thus advancing understanding in combinatorial geometry.

Contribution

It determines the asymptotic behavior of the extremal function for regular simplices in high dimensions and solves a conjecture of Erdős for the case of three-dimensional simplices.

Findings

Asymptotic bounds for $S^k_d(n)$ in fixed dimensions

Exact count of $S^3_d(n)$ for even $d \\geq 6$

Resolution of Erd\H{o}s's conjecture in a stronger form

Abstract

We study the extremal function $S^k_d(n)$, defined as the maximum number of regular $(k-1)$-simplices spanned by $n$ points in $\mathbb{R}^d$. For any fixed $d\geq2k\geq6$, we determine the asymptotic behavior of $S^k_d(n)$ up to a multiplicative constant in the lower-order term. In particular, when $k=3$, we determine the exact value of $S^3_d(n)$, for all even dimensions $d\geq6$ and sufficiently large $n$. This resolves a conjecture of Erd\H{o}s in a stronger form. The proof leverages techniques from hypergraph Tur\'an theory and linear algebra.