Solution
= 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 produces
$$
x_1<x_2<\cdots.
$$
Apply the same theorem separately to the negative integers. If the selected indices satisfy $n_1<n_2<\cdots$, then
$$
y_i=-n_i
$$
forms a monochromatic set with $y_1>y_2>\cdots$. The two monochromatic colours need not agree.