   ISSN 1088-6850(online) ISSN 0002-9947(print)

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
MathSciNet review: 3710649
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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 $|X|=|\mathbb R|$, 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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David Fernández-Bretón
Affiliation: Department of Mathematics, University of Michigan, 2074 East Hall, 530 Church Street, Ann Arbor, Michigan 48109-1043
MR Author ID: 1023869
Email: djfernan@umich.edu

Assaf Rinot
Affiliation: Department of Mathematics, Bar-Ilan University, Ramat-Gan 52900, Israel
MR Author ID: 785097
Email: rinotas@math.biu.ac.il

Keywords: Hindman’s Theorem, commutative cancellative semigroups, strong coloring, Jónsson 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)