Transformations of the transfinite plane

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, if admits a stationary set that does not reflect at inaccessibles, then the classical negative partition relation κ → [κ]2κ 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 ϵ G, there exist x ≠ y in X such that f (x + y) = g.