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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.
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 $.

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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