Ramsey's Theorem for a Class of Categories.
Ramsey's Theorem states that for a sufficiently large set S, and for any splitting of the k-element subsets of S into r classes, there is a subset T [unk] S, [unk]T[unk] = l, such that all k-element subsets of T are in the same class. This paper establishes a theorem for certain categories that generalizes Ramsey's Theorem. In particular, it is strong enough to establish G-C. Rota's conjecture that the vector space analogue to Ramsey's Theorem is true. It also implies the Ramsey theorem for n-parameter sets, which has as corollaries, among others, the theorem of van der Waerden on arithmetic progressions and several results of R. Rado on regularity in systems of linear equations.
Bell Telephone Laboratories, Murray Hill, New Jersey 07974.
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Graham
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Leeb
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Rothschild