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The definition of Ramsey numbers is the following:

Let $R(a,b)$ be a positive number such that every graph of order at least $R(a,b)$ contains either a clique on $a$ vertices or a stable set on $b$ vertices.

I am working on some extension of Ramsey Numbers. While the study has some theoretical interest, it would be important to know the motivation of these numbers. More specifically I am wondering the (theoretical or practical) applications of Ramsey numbers. For instance, are there any solution methodology for a real life problem that uses Ramsey numbers? Or similarly, are there any proofs of some theorems based on Ramsey numbers?

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up vote 21 down vote accepted
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What a great resource! Thanks. – Arman Dec 26 '11 at 18:59
Course in Ramsey Theory and Its "Applications" taught by Gasarch in Spring 2013: – Daniel Apon Feb 11 '13 at 3:11
both urls 404 now =( – vzn Nov 27 '13 at 19:03
re the class notes by apon link in comment. it is currently listed/broken also on this UMD class pg listing classes. maybe moved to here? – vzn Nov 29 '13 at 17:40

In addition to above, Ramsey theory apps by Rosta (Electronic Journal of Combinatorics) also seems to have interesting notes on applications of Ramsey Numbers. Also, I can perceive applications to Genetic Algorithms.

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