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

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2025/iii/paper-130/2/d/solution