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 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.
From: Dingyuan Liu [view email]
[v1]
Sat, 26 Jul 2025 07:30:27 UTC (16 KB)
[v2]
Fri, 3 Jul 2026 12:30:27 UTC (19 KB)
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