Is the challenge to partition the pairs (of natural numbers) such that both partitions are infinite and monochromatic? Is that the hard thing that was proved here?
If so, how about "color red if a = b-1, blue otherwise"? Then the infinite subset (0, 1), (1, 2), (2, 3), ... is monochromatic.
What criterion did my partition there fail to satisfy?
I'm pretty sure that the challenge is to prove that you can for any coloring construct a finitely defined rule for picking the members of the subset which is guaranteed to give you a monochromatic subset.
The question is more about if you can always find such a (finite) rule to partition the set, rather than if you can in a few easily constructed examples.
Ed:
Perhaps I phrased it poorly, but I think the point was to show that you can always construct a predicate, P over a and b, such that P(a, b) is finitely defined (such as "a > 20 and b > 20"), but {(a, b) | P(a, b) is true} is infinite and monochromatic.
Instead of having some cases of colorings where your only option is to construct things of the form "(a = 5 and b = 17) or (a = 3 and b = 47) or ..." where you just list out every pair that matches (in an infinite subset).