The number of regular simplices in higher dimensions

2025-07-26

AI Summary

This paper investigates the asymptotic behavior of $S^k_d(n)$, the maximum number of congruent regular $(k-1)$-simplices determined by $n$ points in $mathbb{R}^d$. Addressing a classical Erdős problem, we establish—*for the first time*—the precise asymptotic order of $S^k_d(n)$ up to the constant factor of the lower-order term when $d ge 2k ge 6$. In particular, for $k=3$ and all even dimensions $d ge 6$, we obtain an exact formula for $S^3_d(n)$ for sufficiently large $n$, thereby *strengthening* Erdős’s conjecture. Methodologically, we introduce a novel synthesis of hypergraph Turán theory and linear-algebraic constraint analysis to construct extremal configurations and characterize their geometric-combinatorial structure. Our results resolve a long-standing open problem in high-dimensional extremal geometry concerning simplex enumeration, and provide a new paradigm for the study of extremal configurations in Euclidean space.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Search and Optimization: Combinatorial Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs

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 $dgeq2kgeq6$, 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 $dgeq6$ and sufficiently large $n$. This resolves a conjecture of Erdős in a stronger form. The proof leverages techniques from hypergraph Turán theory and linear algebra.

Problem

• Determine maximum regular simplices in higher dimensions
• Resolve Erdős conjecture for exact values
• Apply hypergraph Turán theory and linear algebra

Innovation

• Uses hypergraph Turán theory techniques
• Applies linear algebra methods
• Determines asymptotic behavior of simplices