Transformations of the transfinite plane

Joint work with Jing Zhang.

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 $\kappa \nrightarrow [\kappa {]}_{\kappa }^2$ implies that for every Abelian group $(G,+)$ of size $\kappa$, there exists a map $f:G\to G$ such that, for every $X\subseteq G$ of size $\kappa$ and every $g\in G$, there exist $x\ne y$ in $X$ such that $f(x+y)=g$.

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Citation information:

A. Rinot and J. Zhang, Transformations of the transfinite plane, Forum Math. Sigma, 9(e16): 1-25, 2021.