The number of regular simplices in higher dimensions

2025/07/26 by Felix Christian Clemen, Adrian Dumitrescu, Clemen, Felix Christian +3
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paper · pdf · doi:10.48550/arxiv.2507.19841

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 ℝ^d. For any fixed d≥2k≥6, we determine the asymptotic behavior of S^k_d(n) up to a lower-order term. In particular, when k=3, we determine the exact value of S³_d(n), for all even dimensions d≥6 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.

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