OPEN
Let $d\geq 2$ and $n\geq 2$. Let $f_d(n)$ be maximal such that, for any $A\subseteq \mathbb{R}^d$ of size $n$, with diameter $1$, the distance 1 occurs between $f_d(n)$ many pairs of points in $A$. Estimate $f_d(n)$.
Hopf and Pannwitz
[HoPa34] proved $f_2(n)=n$. Heppes
[He56] and Grünbaum-Strasziewicz independently showed that $f_3(n)=2n-2$.
See also [132].
