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Proceedings of the American Mathematical Society

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ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Hindman-like theorems with uncountably many colours and finite monochromatic sets
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by David Fernández-Bretón and Sung Hyup Lee PDF
Proc. Amer. Math. Soc. 148 (2020), 3099-3112 Request permission

Abstract:

A particular case of the Hindman–Galvin–Glazer theorem states that, for every partition of an infinite abelian group $G$ into two cells, there will be an infinite $X\subseteq G$ such that the set of its finite sums $\{x_1+\cdots +x_n \mid n\in \mathbb N\wedge x_1,\ldots ,x_n\in X\text { are distinct}\}$ is monochromatic. It is known that the same statement is false, in a very strong sense, if one attempts to obtain an uncountable (rather than just infinite) $X$. On the other hand, a recent result of Komjáth states that, for partitions into uncountably many cells, it is possible to obtain monochromatic sets of the form $\mathrm {FS}(X)$, for $X$ of some prescribed finite size, when working with sufficiently large Boolean groups. In this paper, we provide a generalization of Komjáth’s result, and we show that, in a sense, this generalization is the strongest possible.
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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
  • Address at time of publication: Kurt Gödel Research Center for Mathematical Logic, University of Vienna, Wäringer Straße 25, 1090 Wien, Austria
  • MR Author ID: 1023869
  • Email: david.fernandez-breton@univie.ac.at
  • Sung Hyup Lee
  • Affiliation: Department of Mathematics, University of California at Berkeley, Berkeley, California 94720
  • Email: sunghlee@berkeley.edu
  • Received by editor(s): July 2, 2018
  • Received by editor(s) in revised form: October 3, 2018
  • Published electronically: March 16, 2020
  • Additional Notes: The first author was partially supported by postdoctoral fellowship number 275049 from Conacyt–Mexico. The second author acknowledges partial support by NSF Grant #DMS-1401384, as part of the University of Michigan Department of Mathematics REU program
  • Communicated by: Heike Mildenberger
  • © Copyright 2020 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 148 (2020), 3099-3112
  • MSC (2010): Primary 03E02; Secondary 03E05, 05D10, 05C55
  • DOI: https://doi.org/10.1090/proc/14649
  • MathSciNet review: 4099796