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Transactions of the American Mathematical Society

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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
DOI: https://doi.org/10.1090/S0002-9947-1971-0284352-8
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.


References [Enhancements On Off] (What's this?)

  • [1] J. Folkman (personal communication).
  • [2] J. Goldman and G.-C. Rota, On the foundations of combinatorial theory. IV: Finite vector spaces and Eulerian generating functions, Studies in Appl. Math., vol. 49, M.I.T. Press, Cambridge, Mass., 1970. MR 0265181 (42:93)
  • [3] R. L. Graham and B. L. Rothschild, Rota's geometric analogue to Ramsey's theorem, Proc. Sympos. Pure Math., vol. 19, Amer. Math. Soc., Providence, R. I., 1971, pp. 101-104.
  • [4] -, Ramsey's theorem for $ n$-dimensional arrays, Bull. Amer. Math. Soc. 75 (1969), 418-422. MR 38 #5638. MR 0237349 (38:5638)
  • [5] A. W. Hales and R. I. Jewett, Regularity and positional games, Trans. Amer. Math. Soc. 106 (1963), 222-229. MR 26 #1265. MR 0143712 (26:1265)
  • [6] B. Huppert, Endliche Gruppen. I, Die Grundlehren der math. Wissenschaften, Band 134, Springer-Verlag, New York, 1967. MR 37 #302. MR 0224703 (37:302)
  • [7] A. Ja. Hinčin, Three pearls of number theory, OGIZ, Moscow, 1948; English transl., Graylock Press, Rochester, N. Y., 1952. MR 11, 83; MR 13, 724. MR 0046372 (13:724b)
  • [8] R. Rado, Note on combinatorial analysis, Proc. London Math. Soc. (2) 48 (1943), 122-160. MR 5, 87. MR 0009007 (5:87a)
  • [9] -, Some partition theorems, Colloq. Math. Soc. János Bolyai, 4, Combinatorial Theory and its Applications, vol. III, North-Holland, Amsterdam, 1970. MR 0297585 (45:6639)
  • [10] F. P. Ramsey, On a problem of formal logic, Proc. London Math. Soc. (2) 30 (1930), 264-286.
  • [11] B. L. Rothschild, A generalization of Ramsey's Theorem and a conjecture of Rota, Doctoral Dissertation, Yale University, New Haven, Conn., 1967.
  • [12] H. J. Ryser, Combinatorial mathematics, Carus Math. Monographs, no. 14, Math. Assoc. Amer.; distributed by, Wiley, New York, 1963. MR 27 #51. MR 0150048 (27:51)
  • [13] J. Sanders, A generalization of a theorem of Schur, Doctoral Dissertation, Yale University, New Haven, Conn., 1968.
  • [14] I. Schur, Über die Kongruenz $ {x^m} + {y^m} \equiv {z^m}\pmod p$, Jber. Deutsch. Math.-Verein. 25 (1916), 114.
  • [15] B. L. van der Waerden, Beweis einer Baudetschen Vermutung, Nieuw Arch. Wisk. 15 (1927), 212-216.

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Additional Information

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

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