i am searching for an on- line paper or a book, or maybe just a paper or a book which consists a proof of finite Ramsey's theorem for sets (not for graphs). i need a combinatorial proof which is not written as a consenquence of the infinite Ramsey's theorem. i know that such paper exits because i have had it few years ago, but i can't remember the name of the paper or the author.
closed as unclear what you're asking by Andrés E. Caicedo, Marco Golla, András Bátkai, Stefan Kohl, Chris Godsil Feb 5 '16 at 18:05
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The proof that Ramsey gives in 'On a problem of formal logic' (Theorem B) does not use the infinite version. In 'Ein kombinatorischer Satz mit Anwendung auf ein logisches Entscheidungsproblem' Skolem gives a simpler proof (in German, though). Since this is the classical Ramsey Theorem, references and proofs can certainly be found in almost any book containing the name 'Ramsey' in its title.
Douglas West - Introduction to graph theory, Theorem 8.3.7. It is Ramsey's theorem for sets, in the form $R(e, p_1, ..., p_k)$. That is, for all sets $N$ with size at least $R(e, p_1, ..., p_k)$, and all colorings of the $e$-subsets of $N$, there exists an i, and a set M of size p_i, such that all the $e$-subsets of $M$ get color $i$. However the actual proof only does $R(e, p_1, p_2)$, which appears to be what you are looking for.