 Solution

Define a six-coloring of ${\mathbb{R}}^n$ byLet $p,q,r$ be the vertices of a unit equilateral triangle and let $o$ be its center. Its circumradius is $1/\sqrt{3}$, and the translation-invariant quadratic identity isSuppose all four points had one color $s$. WriteMultiplying the quadratic identity by $3$ givesThe final expression lies strictly between $-3$ and $3$, whereas $3$ is at distance exactly $3$ from the nearest multiple of $6$. This is impossible. The coloring therefore contains no monochromatic copy of the four-point configuration in any dimension $n\ge 2$.

Solved by gpt-5.6-sol high.