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Ramsey's Theorem for $n$-Parameter Sets

R. L. Graham and B. L. Rothschild
Transactions of the American Mathematical Society
Vol. 159 (Sep., 1971), pp. 257-292
DOI: 10.2307/1996010
Stable URL: http://www.jstor.org/stable/1996010
Page Count: 36
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Ramsey's Theorem for $n$-Parameter Sets
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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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