Hindman and Owings-like theorems without the Axiom of Choice

Abstract

We investigate Hindman- and Owings-type Ramsey-theoretic statements in Zermelo-Fraenkel set theory without the Axiom of Choice, with some occasional extra assumptions (such as the Axiom of Dependent Choice and/or the Axiom of Determinacy). We study several variations of Hindman's theorem on $\mathbb Q$-vector spaces; notably, we show that the uncountable analog of Hindman's theorem fails for the additive group of $\mathbb R$ (under ZF), and for $\mathbb Q$-vector spaces of uncountable dimension (under DC if such dimension is not well-orderable), among other results. In contrast, for Owings-type configurations, we obtain several positive results, especially when assuming AD. These results highlight the interaction between determinacy, algebraic structure, and dimension in the study of infinite Ramsey theory without the Axiom of Choice.