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Transformations of the transfinite plane
Published online by Cambridge University Press: 03 March 2021
Abstract
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 $f(x+y)=g$.
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- 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.
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- © The Author(s), 2021. Published by Cambridge University Press
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