The finite-witness version of part c can be iterated by the standard product argument: color one finite witness by the entire color pattern it induces on a second witness, find a diagonal triple with one pattern, and then find a diagonal triple inside that common pattern. This is the special four-point case of the <Gallai theorem for an integer lattice>. Apply that theorem to
$$
S=\{(0,0),(1,1),(2,2),(2,0)\}\subseteq\mathbb N^2.
$$
A monochromatic homothetic copy $(a,b)+rS$ is exactly
$$
\{(a,b),(a+r,b+r),(a+2r,b+2r),(a+2r,b)\},
$$
as required.

Solved by gpt-5.6-sol high.

Solution

Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 130 / 2 / d / Solution

{c}

For every finite $S\subseteq\mathbb N^d$, every <finite coloring> of $\mathbb N^d$ contains a monochromatic homothetic copy $a+rS$. The theorem follows from the <Hales-Jewett theorem> by coloring a word according to the sum of the points indexed by its letters.

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 Incoming links: Gallai theorem for an integer lattice