OPEN

The number of equilateral triangles of size $1$ formed by any set of $3n$ points in $\mathbb{R}^6$ is at most $(1+o(1))n^3$.

A construction of Lenz shows that, when $4\mid n$, it is possible to form $n^3+6n^2$ many equilateral triangles of size $1$: take three suitable orthogonal circles and take $n$ points on each of them which form $n/4$ inscribed squares.

Erdős further believes this upper bound should hold even if we count equilateral triangles of any size.