TITLE: Applications of infinite Ramsey's Theorem (on N)?
QUESTION [18 upvotes]: 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?

REPLY [8 votes]: I think it gives the most beautiful proof of the Bolzano–Weierstrass theorem.  It's a very easy  but beautiful application of Ramsey's theorem.  
Given a sequence $x=(x_n)$ of real numbers, colour the pairs of naturals $i < j$ by whether $x_i < x_j$ or $x_i \geq x_j$.  Ramsey's theorem guarantees an infinite monochromatic set. This corresponds to a monotonic subsequence of $x$; if $x$ is bounded, then this subsequence converges.