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Hindman and Owings-like theorems without the Axiom of Choice

José A. Guzmán-Vega, David J. Fernández Bretón, Eliseo Sarmiento Rosales

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.

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 -vector spaces; notably, we show that the uncountable analog of Hindman's theorem fails for the additive group of (under ZF), and for -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.

Paper Structure

This paper contains 10 sections, 15 theorems, 15 equations.

Table of Contents

  1. Introduction
  2. Hindman-like results
  3. Finite-sums notation and combinatorial tools
  4. Negative results
  5. Positive finite configurations from partition properties
  6. Owings-like results
  7. Pairwise-sum notation and background
  8. Positive results on $\mathbb{R}$ from regularity
  9. Algebraic limitations and large-dimensional positives
  10. Conclusions

Key Result

Theorem 2.3

For all finite $\kappa,\theta$ there exists an $S\in\mathbb{N}$ such that, for every Abelian group $G$ with $|G|\geq S$, $G\to(\kappa)_{\theta}^{\operatorname{FS}}$.

Theorems & Definitions (35)

  • Definition 2.1
  • Remark 2.2
  • Theorem 2.3
  • proof
  • Theorem 2.4
  • proof
  • Remark 2.5
  • Definition 2.6
  • Lemma 2.7: $\mathnormal{\mathsf{DC}}$
  • proof
  • ...and 25 more