Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 130 / 2 / i

This is always true. Pull the given coloring of ${\mathbb{N}}^2$ back along the dilationApply the Gallai theorem for an integer lattice to the four vertices of the unit square. The image of the resulting homothetic square has side length $2q$, which is even, and all four vertices have one color.