Let $T_6^u(n)$ count the maximal number of equilateral triangles of size $1$ formed by $n$ points in $\mathbb{R}^6$. The following construction (which
[Er94b] attributes to Lenz, but appears in an earlier paper of Erdős and Purdy
[ErPu75]) shows that\[T_6^u(n)\geq \frac{n^3}{27}-O(n^2):\]take three suitable orthogonal circles and take $n/3$ points on each of them which form $n/4$ inscribed squares.
Erdős believed this conjectured upper bound should hold even if we count equilateral triangles of any size.
This problem was solved in a strong form by Clemen, Dumitrescu, and Liu
[CDL25b], who in fact prove that if $T_6(n)$ is the maximal number of equilateral triangles of any size formed by $n$ points in $\mathbb{R}^6$ then\[T_6(n)=(\tfrac{1}{27}+o(1))n^3.\]Indeed, they give an exact formula for $T_d(n)$ for all even $d\geq 6$ (and sufficiently large $n$).
See also
[1086].
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This page was last edited 16 October 2025. View history
When referring to this problem, please use the original sources of Erdős. If you wish to acknowledge this website, the recommended citation format is:
T. F. Bloom, Erdős Problem #755, https://www.erdosproblems.com/755, accessed 2026-04-11
