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On the Rainbow Ramsey theorem and the Canonical Ramsey Theorem for pairs without AC

Amitayu Banerjee, Alexa Gopaulsingh, Zalán Molnár

Abstract

In set theory without the Axiom of Choice, we study the set-theoretic strength of a generalized version of the Rainbow Ramsey theorem and the Canonical Ramsey Theorem for pairs introduced by Erdős and Rado, concerning their interrelation with several weak choice forms.

On the Rainbow Ramsey theorem and the Canonical Ramsey Theorem for pairs without AC

Abstract

In set theory without the Axiom of Choice, we study the set-theoretic strength of a generalized version of the Rainbow Ramsey theorem and the Canonical Ramsey Theorem for pairs introduced by Erdős and Rado, concerning their interrelation with several weak choice forms.
Paper Structure (13 sections, 18 theorems, 5 equations, 1 figure)

This paper contains 13 sections, 18 theorems, 5 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Results.
  3. Amorphous sets and Ramsey type theorems
  4. Diagram of results
  5. Basics
  6. Permutation models
  7. Positive results and known results
  8. Positive results
  9. Rainbow Ramsey theorem, CRT, and weak choice forms
  10. Results in the Basic Cohen Model
  11. Amorphous sets and Ramsey type theorems
  12. Concluding Remarks
  13. Questions

Key Result

Lemma 2.4

An element $x$ of $\mathcal{N}_{\mathcal{I}}$ is well-orderable in $\mathcal{N}_{\mathcal{I}}$ if and only if fix$_{\mathcal{G}}(x)\in \mathcal{F}_{\mathcal{I}}$ where $\mathcal{F}_{\mathcal{I}}$ is the normal filter generated by the filter base $\{$ fix$_{\mathcal{G}}(E): E\in\mathcal{I}\}$ (cf. Je

Figures (1)

  • Figure 1: Implications/non-implications between the principles.

Theorems & Definitions (58)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Lemma 2.4
  • Lemma 2.5
  • Proposition 3.2
  • proof
  • Proposition 3.3
  • proof
  • claim 3.4
  • ...and 48 more