Finite Ramsey's theorem is a very important combinatorial tool that is often used in mathematics. The infinite version of Ramsey's theorem (Ramsey's theorem for colorings of tuples of natural numbers) also seems to be a very basic and powerful tool but it is apparently not as widely used.

I searched in the literature for applications of infinite Ramsey's theorem and only found

• straight forward generalization of statements that follow from finite Ramsey's theorem (example: Erdos-Szekeres ~> every infinite sequence of reals contains a monotonic subsequence) and some other basic combinatorial applications,
• Ramsey factorization for \omega-words,
• the original applications of Ramsey to Logic.

Where else is infinite Ramsey's theorem used? Especially are there applications to analysis?