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Strong failures of higher analogs of Hindman's theorem


Authors: David Fernández-Bretón and Assaf Rinot
Journal: Trans. Amer. Math. Soc. 369 (2017), 8939-8966
MSC (2010): Primary 03E02; Secondary 03E75, 03E35
DOI: https://doi.org/10.1090/tran/7131
Published electronically: May 31, 2017
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Abstract: We show that various analogs of Hindman's theorem fail in a strong sense when one attempts to obtain uncountable monochromatic sets:

Theorem 1. There exists a colouring $ c:\mathbb{R}\rightarrow \mathbb{Q}$, such that for every $ X\subseteq \mathbb{R}$ with $ \vert X\vert=\vert\mathbb{R}\vert$, and every colour $ \gamma \in \mathbb{Q}$, there are two distinct elements $ x_0,x_1$ of $ X$ for which $ c(x_0+x_1)=\gamma $. This forms a simultaneous generalization of a theorem of Hindman, Leader and Strauss and a theorem of Galvin and Shelah.

Theorem 2. For every abelian group $ G$, there exists a colouring $ c:G\rightarrow \mathbb{Q}$ such that for every uncountable $ X\subseteq G$ and every colour $ \gamma $, for some large enough integer $ n$, there are pairwise distinct elements $ x_0,\ldots ,x_n$ of $ X$ such that $ c(x_0+\cdots +x_n)=\gamma $. In addition, it is consistent that the preceding statement remains valid even after enlarging the set of colours from $ \mathbb{Q}$ to $ \mathbb{R}$.

Theorem 3. Let $ \circledast _\kappa $ assert that for every abelian group $ G$ of cardinality $ \kappa $, there exists a colouring $ c:G\rightarrow G$ such that for every positive integer $ n$, every $ X_0,\ldots ,X_n \in [G]^\kappa $, and every $ \gamma \in G$, there are $ x_0\in X_0,\ldots , x_n\in X_n$ such that $ c(x_0+\cdots +x_n)=\gamma $. Then $ \circledast _\kappa $ holds for unboundedly many uncountable cardinals $ \kappa $, and it is consistent that $ \circledast _\kappa $ holds for all regular uncountable cardinals $ \kappa $.

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

  • [1] Andreas Blass, Weak partition relations, Proc. Amer. Math. Soc. 35 (1972), 249-253. MR 0297576, https://doi.org/10.2307/2038480
  • [2] Andreas Blass and Neil Hindman, On strongly summable ultrafilters and union ultrafilters, Trans. Amer. Math. Soc. 304 (1987), no. 1, 83-97. MR 906807, https://doi.org/10.2307/2000705
  • [3] Robert Bonnet and Saharon Shelah, Narrow Boolean algebras, Ann. Pure Appl. Logic 28 (1985), no. 1, 1-12. MR 776283, https://doi.org/10.1016/0168-0072(85)90028-4
  • [4] Todd Eisworth, Getting more colors II, J. Symbolic Logic 78 (2013), no. 1, 17-38. MR 3087059, https://doi.org/10.2178/jsl.7801020
  • [5] Paul Erdős, András Hajnal, Attila Máté, and Richard Rado, Combinatorial set theory: partition relations for cardinals, Studies in Logic and the Foundations of Mathematics, vol. 106, North-Holland Publishing Co., Amsterdam, 1984. MR 795592
  • [6] David J. Fernández-Bretón, Hindman's theorem is only a countable phenomenon, to appear in Order, DOI 10.1007/s11083-016-9419-7; available online at https://protect-us.mimecast.com/s/dqpgBRHaekh8?domain=link.springer.com.
  • [7] Matthew Foreman and W. Hugh Woodin, The generalized continuum hypothesis can fail everywhere, Ann. of Math. (2) 133 (1991), no. 1, 1-35. MR 1087344, https://doi.org/10.2307/2944324
  • [8] László Fuchs, Infinite abelian groups. Vol. I, Pure and Applied Mathematics, Vol. 36, Academic Press, New York-London, 1970. MR 0255673
  • [9] Fred Galvin, Chain conditions and products, Fund. Math. 108 (1980), no. 1, 33-48. MR 585558
  • [10] Fred Galvin and Saharon Shelah, Some counterexamples in the partition calculus, J. Combinatorial Theory Ser. A 15 (1973), 167-174. MR 0329900
  • [11] Kurt Gödel, The consistency of the continuum hypothesis, Annals of Mathematics Studies, no. 3, Princeton University Press, Princeton, N. J., 1940. MR 0002514
  • [12] Neil Hindman, Finite sums from sequences within cells of a partition of $ N$, J. Combinatorial Theory Ser. A 17 (1974), 1-11. MR 0349574
  • [13] Neil Hindman, Leader Imre, and Dona Strauss, Pairwise sums in colourings of the reals, to appear in Halin Memorial Volume (2015).
  • [14] Neil Hindman and Dona Strauss, Algebra in the Stone-Čech compactification: Theory and applications, De Gruyter Textbook, Walter de Gruyter & Co., Berlin, 2012. Second revised and extended edition [of MR1642231]. MR 2893605
  • [15] Thomas Jech, Set theory, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2003. The third millennium edition, revised and expanded. MR 1940513
  • [16] R. Björn Jensen, The fine structure of the constructible hierarchy, Ann. Math. Logic 4 (1972), 229-308; erratum, ibid. 4 (1972), 443. MR 0309729, https://doi.org/10.1016/0003-4843(72)90001-0
  • [17] Akihiro Kanamori, The higher infinite: Large cardinals in set theory from their beginnings, 2nd ed., Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2009. Paperback reprint of the 2003 edition. MR 2731169
  • [18] Péter Komjáth, A certain 2-coloring of the reals, Real Anal. Exchange 41 (2016), no. 1, 227-231. MR 3511943
  • [19] Keith R. Milliken, Hindman's theorem and groups, J. Combin. Theory Ser. A 25 (1978), no. 2, 174-180. MR 0505558, https://doi.org/10.1016/0097-3165(78)90079-1
  • [20] Justin Tatch Moore, A solution to the $ L$ space problem, J. Amer. Math. Soc. 19 (2006), no. 3, 717-736. MR 2220104, https://doi.org/10.1090/S0894-0347-05-00517-5
  • [21] Yinhe Peng and Liuzhen Wu, A Lindelöf topological group with non-Lindelöf square, preprint (2014).
  • [22] Assaf Rinot, A topological reflection principle equivalent to Shelah's strong hypothesis, Proc. Amer. Math. Soc. 136 (2008), no. 12, 4413-4416. MR 2431057, https://doi.org/10.1090/S0002-9939-08-09411-2
  • [23] Assaf Rinot, Transforming rectangles into squares, with applications to strong colorings, Adv. Math. 231 (2012), no. 2, 1085-1099. MR 2955203, https://doi.org/10.1016/j.aim.2012.06.013
  • [24] Assaf Rinot, Chain conditions of products, and weakly compact cardinals, Bull. Symb. Log. 20 (2014), no. 3, 293-314. MR 3271280, https://doi.org/10.1017/bsl.2014.24
  • [25] Assaf Rinot, Complicated colorings, Math. Res. Lett. 21 (2014), no. 6, 1367-1388. MR 3335852, https://doi.org/10.4310/MRL.2014.v21.n6.a9
  • [26] Frederick Rowbottom, Some strong axioms of infinity incompatible with the axiom of constructibility, Ann. Math. Logic 3 (1971), no. 1, 1-44. MR 0323572, https://doi.org/10.1016/0003-4843(71)90009-X
  • [27] Saharon Shelah, Was Sierpiński right? I, Israel J. Math. 62 (1988), no. 3, 355-380. MR 955139, https://doi.org/10.1007/BF02783304
  • [28] Saharon Shelah, Cardinal arithmetic, Oxford Logic Guides, vol. 29, The Clarendon Press, Oxford University Press, New York, 1994. MR 1318912
  • [29] Richard A. Shore, Square bracket partition relations in $ L$, Fund. Math. 84 (1974), no. 2, 101-106. MR 0371662
  • [30] Waclaw Sierpiński, Sur un problème de la théorie des relations, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (2) 2 (1933), no. 3, 285-287 (French). MR 1556708
  • [31] Daniel Soukup and William Weiss, Pairwise sums in colourings of the reals, preprint, available online at http://www.renyi.hu/~dsoukup/finset_colouring.pdf (2016).
  • [32] Stevo Todorčević, Remarks on chain conditions in products, Compositio Math. 55 (1985), no. 3, 295-302. MR 799818
  • [33] Stevo Todorčević, Partitioning pairs of countable ordinals, Acta Math. 159 (1987), no. 3-4, 261-294. MR 908147, https://doi.org/10.1007/BF02392561
  • [34] Stevo Todorčević, Partition problems in topology, Contemporary Mathematics, vol. 84, American Mathematical Society, Providence, RI, 1989. MR 980949

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Additional Information

David Fernández-Bretón
Affiliation: Department of Mathematics, University of Michigan, 2074 East Hall, 530 Church Street, Ann Arbor, Michigan 48109-1043
Email: djfernan@umich.edu

Assaf Rinot
Affiliation: Department of Mathematics, Bar-Ilan University, Ramat-Gan 52900, Israel
Email: rinotas@math.biu.ac.il

DOI: https://doi.org/10.1090/tran/7131
Keywords: Hindman's Theorem, commutative cancellative semigroups, strong coloring, J\'onsson cardinal
Received by editor(s): September 23, 2016
Received by editor(s) in revised form: November 14, 2016
Published electronically: May 31, 2017
Additional Notes: The first author was partially supported by Postdoctoral Fellowship number 263820/275049 from the Consejo Nacional de Ciencia y Tecnología (CONACyT), Mexico. The second author was partially supported by the Israel Science Foundation (grant $#$1630/14).
Dedicated: This paper is dedicated to the memory of András Hajnal (1931–2016)
Article copyright: © Copyright 2017 American Mathematical Society

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