= Successive thinning proof of the infinite Ramsey theorem
{title2=$\mathbb N\longrightarrow(\mathbb N)^r_k$}
For finitely coloured $(r+1)$-element subsets, choose successive least points and use the inductive $r$-set theorem to thin the remaining reservoir after each choice. Every tuple with that least point then has one assigned colour. The <infinite pigeonhole principle> retains infinitely many least points with the same assigned colour, giving an infinite homogeneous set. The $r=1$ base case is the same pigeonhole principle. This proves the <Ramsey theorem for r-sets> for all finite $r$.
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