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Transformations of the transfinite plane

Part of:Set theory

Published online by Cambridge University Press:  03 March 2021

Assaf Rinot and
Jing Zhang
Assaf Rinot
Affiliation:
Department of Mathematics, Bar-Ilan University, Ramat-Gan 5290002, Israel; E-mail: rinotas@math.biu.ac.il, jingzhangjz13@gmail.com.
Jing Zhang
Affiliation:
Department of Mathematics, Bar-Ilan University, Ramat-Gan 5290002, Israel; E-mail: rinotas@math.biu.ac.il, jingzhangjz13@gmail.com.

Abstract

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We study the existence of transformations of the transfinite plane that allow one to reduce Ramsey-theoretic statements concerning uncountable Abelian groups into classical partition relations for uncountable cardinals.

To exemplify: we prove that for every inaccessible cardinal $\kappa $, if $\kappa $ admits a stationary set that does not reflect at inaccessibles, then the classical negative partition relation $\kappa \nrightarrow [\kappa ]^2_\kappa $ implies that for every Abelian group $(G,+)$ of size $\kappa $, there exists a map $f:G\rightarrow G$ such that for every $X\subseteq G$ of size $\kappa $ and every $g\in G$, there exist $x\neq y$ in X such that $f(x+y)=g$.

MSC classification

Primary:03E02: Partition relations
Secondary:03E35: Consistency and independence results

Information

Type
Foundations
Information
Forum of Mathematics, Sigma , Volume 9 , 2021, e16
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Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/ ), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s), 2021. Published by Cambridge University Press

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