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Hindman’s Theorem is only a Countable Phenomenon

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We pursue the idea of generalizing Hindman’s Theorem to uncountable cardinalities, by analogy with the way in which Ramsey’s Theorem can be generalized to weakly compact cardinals. But unlike Ramsey’s Theorem, the outcome of this paper is that the natural generalizations of Hindman’s Theorem proposed here tend to fail at all uncountable cardinals.

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  1. Baumgartner, J.: A Short Proof of Hindman’s Theorem. J. Combin. Theory Ser. A. 17, 384–386 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  2. Carlson, T.: Some unifying principles in Ramsey theory. Discret. Math. 68, 117–169 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  3. Fernández Bretón, D.: Every Strongly Summable Ultrafilter on \(\bigoplus \mathbb {Z}_{2}\) is Sparse. New York J. Math. 19, 117–129 (2013)

  4. Fernández-Bretón, D., Rinot, A.: Strong failures of higher analogs of Hindman’s theorem. To appear in Transactions of the American Mathematical Society. arXiv:1608.01512

  5. Furstenberg, H., Katznelson, Y.: Idempotents in compact semigroups and Ramsey theory. Israel J. Math. 68, 257–270 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  6. Guzmán González, O., Hrušák, M.: (personal communication)

  7. Hindman, N.: Finite Sums from Sequences Within Cells of a Partition of N. J. Combin. Theory Ser. A. 17, 1–11 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  8. Hindman, N., Strauss, D.: Algebra in the Stone-Cech Compactification, 2nd edn. Walter de Gruyter, Berlin (2012)

    MATH  Google Scholar 

  9. Just, W., Weese, M.: Discovering Modern Set Theory II. Set-Theoretic Tools for Every Mathematician. Graduate Studies in Mathematics Vol. 18. American Mathematical Society (1995)

  10. Kunen, K.: Set Theory. An introduction to independence proofs, Studies in Logic and the Foundations of Mathematics, vol. 102. North Holland (1980)

  11. Milliken, K.R.: Hindman’s theorem and groups. J. Combin. Theory Ser. A 25, 174–180 (1978)

    Article  MathSciNet  Google Scholar 

  12. Todorcevic, S.: Introduction to Ramsey Spaces. Annals of Mathematics Studies. Princeton University Press (2010)

  13. Tsaban, B.: Algebra, selections, and additive Ramsey Theory. Preprint arXiv:1407.7437

  14. Zheng, Y.Y.: Selective ultrafilters on FIN. Unpublished note (available online at

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Correspondence to David J. Fernández-Bretón.

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The author was partially supported by postdoctoral fellowship number 263820 from the Consejo Nacional de Ciencia y Tecnología (Conacyt), Mexico.

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Fernández-Bretón, D.J. Hindman’s Theorem is only a Countable Phenomenon. Order 35, 83–91 (2018).

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