Exercise in set theory

Question

Let $\mathbb{N}^{[2]}$ be the set of all sets with two elements in $\mathbb{N}$, and let $\mathbb{N}^{[2]}=A\cup B$. Prove that there is an infinite set $M\subseteq \mathbb{N}$ such that either $M^{[2]}\subseteq A$ or $M^{[2]}\subseteq B$. Any ideas?

I found a hint telling to use the well-ordering principle

Answer 1

This is a classical theorem of Ramsey. There are many proofs. See for example Wikipedia.