The number of regular simplices in higher dimensions
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866908468618723328 |
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| author | Clemen, Felix Christian Dumitrescu, Adrian Liu, Dingyuan |
| author_facet | Clemen, Felix Christian Dumitrescu, Adrian Liu, Dingyuan |
| contents | 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 Erdős in a stronger form. The proof leverages techniques from hypergraph Turán theory and linear algebra. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_19841 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The number of regular simplices in higher dimensions Clemen, Felix Christian Dumitrescu, Adrian Liu, Dingyuan Combinatorics Computational Geometry 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 Erdős in a stronger form. The proof leverages techniques from hypergraph Turán theory and linear algebra. |
| title | The number of regular simplices in higher dimensions |
| topic | Combinatorics Computational Geometry |
| url | https://arxiv.org/abs/2507.19841 |