Hindman and Owings-like theorems without the Axiom of Choice

TL;DR

This paper explores Hindman- and Owings-type Ramsey statements within ZF set theory, revealing failures and successes depending on algebraic and set-theoretic assumptions, especially regarding uncountable structures.

Contribution

It demonstrates the failure of uncountable Hindman-type theorems in ZF and the positive results for Owings-type configurations under determinacy assumptions.

Findings

Uncountable Hindman's theorem fails for al R and uncountable al Q-vector spaces under ZF.

Owings-type configurations have positive results under AD assumptions.

The results show the interplay between determinacy, algebra, and dimension in Ramsey theory without 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.