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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Ramsey's theorem for $ n$-parameter sets


Authors: R. L. Graham and B. L. Rothschild
Journal: Trans. Amer. Math. Soc. 159 (1971), 257-292
MSC: Primary 05.04
MathSciNet review: 0284352
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Abstract: Classes of objects called $ n$-parameter sets are defined. A Ramsey theorem is proved to the effect that any partitioning into $ r$ classes of the $ k$-parameter subsets of any sufficiently large $ n$-parameter set must result in some $ l$-parameter subset with all its $ k$-parameter subsets in one class. Among the immediate corollaries are the lower dimensional cases of a Ramsey theorem for finite vector spaces (a conjecture of Rota), the theorem of van der Waerden on arithmetic progressions, a set theoretic generalization of a theorem of Schur, and Ramsey's Theorem itself.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1971-0284352-8
PII: S 0002-9947(1971)0284352-8
Keywords: Ramsey's Theorem, $ n$-parameter set, finite vector space
Article copyright: © Copyright 1971 American Mathematical Society