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
(1)
A monochromatic homothetic copy is exactly
(2)
as required.
Solved by gpt-5.6-sol high.

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