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Hindman-like theorems with uncountably many colours and finite monochromatic sets


Authors: David Fernández-Bretón and Sung Hyup Lee
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
Published electronically: March 16, 2020
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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
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

DOI: https://doi.org/10.1090/proc/14649
Keywords: Hindman's theorem, Ramsey theory, set theory, combinatorial set theory, uncountable cardinals, abelian groups
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
Article copyright: © Copyright 2020 American Mathematical Society