Abstract
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.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Baumgartner, J.: A Short Proof of Hindman’s Theorem. J. Combin. Theory Ser. A. 17, 384–386 (1974)
Carlson, T.: Some unifying principles in Ramsey theory. Discret. Math. 68, 117–169 (1988)
Fernández Bretón, D.: Every Strongly Summable Ultrafilter on \(\bigoplus \mathbb {Z}_{2}\) is Sparse. New York J. Math. 19, 117–129 (2013)
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
Furstenberg, H., Katznelson, Y.: Idempotents in compact semigroups and Ramsey theory. Israel J. Math. 68, 257–270 (1989)
Guzmán González, O., Hrušák, M.: (personal communication)
Hindman, N.: Finite Sums from Sequences Within Cells of a Partition of N. J. Combin. Theory Ser. A. 17, 1–11 (1974)
Hindman, N., Strauss, D.: Algebra in the Stone-Cech Compactification, 2nd edn. Walter de Gruyter, Berlin (2012)
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)
Kunen, K.: Set Theory. An introduction to independence proofs, Studies in Logic and the Foundations of Mathematics, vol. 102. North Holland (1980)
Milliken, K.R.: Hindman’s theorem and groups. J. Combin. Theory Ser. A 25, 174–180 (1978)
Todorcevic, S.: Introduction to Ramsey Spaces. Annals of Mathematics Studies. Princeton University Press (2010)
Tsaban, B.: Algebra, selections, and additive Ramsey Theory. Preprint arXiv:1407.7437
Zheng, Y.Y.: Selective ultrafilters on FIN. Unpublished note (available online at http://www.math.toronto.edu/yyz22/2selfin.pdf)
Author information
Authors and Affiliations
Corresponding author
Additional information
The author was partially supported by postdoctoral fellowship number 263820 from the Consejo Nacional de Ciencia y Tecnología (Conacyt), Mexico.
Rights and permissions
About this article
Cite this article
Fernández-Bretón, D.J. Hindman’s Theorem is only a Countable Phenomenon. Order 35, 83–91 (2018). https://doi.org/10.1007/s11083-016-9419-7
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11083-016-9419-7