Define a six-coloring of $\mathbb R^n$ by
$$
c(x)=\left\lfloor3\lVert x\rVert^2\right\rfloor\pmod6.
$$
Let $p,q,r$ be the vertices of a unit <equilateral triangle> and let $o$ be its center. Its circumradius is $1/\sqrt3$, and the translation-invariant quadratic identity is
$$
\lVert p\rVert^2+\lVert q\rVert^2+\lVert r\rVert^2-3\lVert o\rVert^2=1.
$$
Suppose all four points had one color $s$. Write
$$
\left\lfloor3\lVert x\rVert^2\right\rfloor=6m_x+s,
\qquad
\theta_x=\left\{3\lVert x\rVert^2\right\}∈[0,1).
$$
Multiplying the quadratic identity by $3$ gives
$$
3=6(m_p+m_q+m_r-3m_o)
+\theta_p+\theta_q+\theta_r-3\theta_o.
$$
The 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 $n\geq2$.

Solved by gpt-5.6-sol high.

Solution

Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 130 / 4 / d / Solution

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An equilateral triangle has three equal side lengths.


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