The number of regular simplices in higher dimensions

We study the extremal function $S_d^k(n)$, defined as the maximum number of regular $(k-1)$-simplices spanned by $n$ points in ${\mathbb{R}}^d$. For any fixed $d\ge 2k\ge 6$, we determine the asymptotic behavior of $S_d^k(n)$ up to a multiplicative constant in the lower-order term. In particular, when $k=3$, we determine the exact value of $S_d^3(n)$, for all even dimensions $d\ge 6$ 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.