Abstract
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 $f(x+y)=g$ .
Highlights
Ramsey’s theorem [Ram30] asserts that every infinite graph contains an infinite induced subgraph that is either a clique or an anti-clique
For every function : [N]2 → 2, there exists an infinite ⊆ N that is monochromatic in the sense that, for some ∈ 2, (, ) = for every pair < of elements of
A strengthening of Ramsey’s theorem due to Hindman [Hin74] concerns the additive structure (N, +) and asserts that for every partition : N → 2, there exists an infinite ⊆ N that is monochromatic in the sense that, for some ∈ 2, for every finite increasing sequence 0 < · · · < of elements of, ( 0 + · · · + ) =
Summary
Ramsey’s theorem [Ram30] asserts that every infinite graph contains an infinite induced subgraph that is either a clique or an anti-clique. A strengthening of Ramsey’s theorem due to Hindman [Hin74] concerns the additive structure (N, +) and asserts that for every partition : N → 2, there exists an infinite ⊆ N that is monochromatic in the sense that, for some ∈ 2, for every finite increasing sequence 0 < · · · < of elements of , ( 0 + · · · + ) =. A natural generalization of Ramsey’s and Hindman’s theorems would assert that in any 2-partition of an uncountable structure, there must exist an uncountable monochromatic subset. For any ∈ Reg( ), the existence of a nonreflecting stationary subset of Pl1 ( , 1, +)
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