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The classical problem in Ramsey theory is the party problem, which asks the minimum number of guests that must be invited so that at least m will know each other (i.e., there exists a clique of order m) or at least n will not know each other (i.e., there exists an independent set of order n). Here, R(m,n) is called a Ramsey number.

See on Wolfram MathWorld or wikipedia.

There seems to be a slew of research in the field. What are some real world applications? For example, I read in one place that circuit/network routing would be a good one. But how why is that? Does anyone know of any other real world applications?

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marked as duplicate by Sasho Nikolov, Tsuyoshi Ito, Kaveh Feb 10 '13 at 6:26

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

    
ps: reminder that cstheory is a Q&A site for research-level questions in theoretical computer science (TCS). Please see the FAQ for more information on what is meant by this. –  Kaveh Feb 10 '13 at 7:01

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A course I'm currently taking from Bill Gasarch is called: Ramsey Theory and Its 'Applications' -- the quotes are in the official course title. Interpret that as you wish. >_>

Here is a link to the course website (with lecture notes): http://www.cs.umd.edu/~gasarch/858/S13/S13.html

Here is a link to Bill's "Apps of Ramsey Theory" page: http://www.cs.umd.edu/~gasarch/ramsey/ramsey.html

The latter link is very likely the most complete collection of information for the question you're asking.

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Daniel, would you mind moving this answer to the other question? –  Kaveh Feb 10 '13 at 6:28
    
Sure -- And in fact, checking the other question, there's already a link to the Apps of Ramsey Theory page. I'll add the course link to it. –  Daniel Apon Feb 11 '13 at 3:11

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