Ramsey theory in noncommutative semigroups

Bergelson, Vitaly; Hindman, Neil

doi:10.1090/S0002-9947-1992-1069744-5

Ramsey theory in noncommutative semigroups
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by Vitaly Bergelson and Neil Hindman

Trans. Amer. Math. Soc. 330 (1992), 433-446

DOI: https://doi.org/10.1090/S0002-9947-1992-1069744-5

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Abstract:

By utilizing ultrafilters we give a general version of the Central Sets Theorem [$6$, Proposition 8.21]. This enables us to derive noncommutative versions of van der Waerden’s Theorem and several of its generalizations. We also derive some standard results, including the Hales-Jewett Theorem.

References

Vitaly Bergelson, Hillel Furstenberg, Neil Hindman, and Yitzhak Katznelson, An algebraic proof of van der Waerden’s theorem, Enseign. Math. (2) 35 (1989), no. 3-4, 209–215. MR 1039944
Vitaly Bergelson and Neil Hindman, Nonmetrizable topological dynamics and Ramsey theory, Trans. Amer. Math. Soc. 320 (1990), no. 1, 293–320. MR 982232, DOI 10.1090/S0002-9947-1990-0982232-5
Timothy J. Carlson and Stephen G. Simpson, A dual form of Ramsey’s theorem, Adv. in Math. 53 (1984), no. 3, 265–290. MR 753869, DOI 10.1016/0001-8708(84)90026-4
Walter Deuber, Partitionen und lineare Gleichungssysteme, Math. Z. 133 (1973), 109–123 (German). MR 325406, DOI 10.1007/BF01237897
Robert Ellis, Lectures on topological dynamics, W. A. Benjamin, Inc., New York, 1969. MR 0267561
H. Furstenberg, Recurrence in ergodic theory and combinatorial number theory, Princeton University Press, Princeton, N.J., 1981. M. B. Porter Lectures. MR 603625
H. Furstenberg and B. Weiss, Topological dynamics and combinatorial number theory, J. Analyse Math. 34 (1978), 61–85 (1979). MR 531271, DOI 10.1007/BF02790008
S. Glasner, Divisible properties and the Stone-Čech compactification, Canadian J. Math. 32 (1980), no. 4, 993–1007. MR 590662, DOI 10.4153/CJM-1980-077-x
R. L. Graham, K. Leeb, and B. L. Rothschild, Ramsey’s theorem for a class of categories, Proc. Nat. Acad. Sci. U.S.A. 69 (1972), 119–120. MR 306009, DOI 10.1073/pnas.69.1.119
A. W. Hales and R. I. Jewett, Regularity and positional games, Trans. Amer. Math. Soc. 106 (1963), 222–229. MR 143712, DOI 10.1090/S0002-9947-1963-0143712-1
Neil Hindman, Ultrafilters and combinatorial number theory, Number theory, Carbondale 1979 (Proc. Southern Illinois Conf., Southern Illinois Univ., Carbondale, Ill., 1979) Lecture Notes in Math., vol. 751, Springer, Berlin, 1979, pp. 119–184. MR 564927

Ultrafilters and Ramsey Theory—an update

A. Yu. Ol′shanskiĭ, Groups of bounded period with subgroups of prime order, Algebra i Logika 21 (1982), no. 5, 553–618 (Russian). MR 721048
Richard Rado, Studien zur Kombinatorik, Math. Z. 36 (1933), no. 1, 424–470 (German). MR 1545354, DOI 10.1007/BF01188632

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