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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

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


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:
  • Sung Hyup Lee
  • Affiliation: Department of Mathematics, University of California at Berkeley, Berkeley, California 94720
  • Email:
  • 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:
  • MathSciNet review: 4099796