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?



Applications of infinite Ramsey's Theorem (on N)?
The following fact has been called "Ramsey's Theorem for Analysts" by H. P. Rosenthal.



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> **Theorem.** Let $(a\_{i,j})\_{i,j=0}^\infty$ be an infinite matrix of real numbers such that $a\_i = {\displaystyle\lim\_{j\to\infty} a\_{i,j}}$ exists for each $i$ and $a = {\displaystyle\lim\_{i\to\infty} a\_i}$ exists too. Then there is an infinite sequence $k(0) < k(1) < k(2) < \cdots$ such that $a = {\displaystyle\lim\_{i<j} a\_{k(i),k(j)}}$.
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The last limit means that for every $\varepsilon > 0$ there is an $n$ such that $n < i < j$ implies $|a-a\_{k(i),k(j)}| < \varepsilon$. When the matrix is symmetric and ${\displaystyle\lim\_{i\to\infty} a\_{k(i),k(i)}} = a$ too, this is just an ordinary double limit. 


The proof is a straightforward applications of the two-dimensional Ramsey's Theorem. The obvious higher dimensional generalizations are also true and they can be established in the same way using the corresponding higher dimensional Ramsey's Theorem. These are used to construct "spreading models" in Banach Space Theory.


