 Solution

Restrict the given finite edge-colouring to the positive integers inside $\mathbb{Q}$. The infinite Ramsey theorem gives an infinite monochromatic subset; listing it in increasing order producesApply the same theorem separately to the negative integers. If the selected indices satisfy $n_1<n_2<\cdots$, thenforms a monochromatic set with $y_1>y_2>\cdots$. The two monochromatic colours need not agree.

Solved by gpt-5.6-sol high.