The number of regular simplices in higher dimensions

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.