Transformations of the transfinite plane

Abstract

We study the existence of transformations of the transfinite plane that allow to reduce Ramsey-theoretic statements concerning uncountable Abelian groups into classic partition relations for uncountable cardinals.To exemplify: we prove that for every inaccessible cardinal κ, if κ admits a stationary set that does not reflect at inaccessibles, then the classic negative partition relation κ[κ] 2 κ implies that for every Abelian group (G, +) of size κ, there exists a map f : G → G such that, for every X ⊆ G of size κ and every g ∈ G, there exist x = y in X such that f (x + y) = g.