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Three forms of the Erdős-Dushnik-Miller Theorem

Paul Howard, Eleftherios Tachtsis

TL;DR

The paper advances the understanding of the Erdős-Dushnik-Miller theorem in the absence of the axiom of choice by distinguishing three inequivalent EDM forms: $\mathsf{EDM_{DM}}$, $\mathsf{EDM_{BG}}$, and $\mathsf{EDM_{T}}$. It establishes a compact trichotomy of EDM forms in the ZF weak-choice hierarchy, and uses Fraenkel-Mostowski models to separate these forms from various weak choice principles, mapping where EDM variants sit relative to PC and AC. The results include proving EDM$_{DM}$ is equivalent to AC, EDM$_{BG}$ and EDM$_{T}$ relations in several FM models, and identifying models where EDM variants withstand or fail under symmetry constraints. The work provides a detailed landscape of how EDM interacts with PC inequalities, UT, and AC in multiple FM constructions, yielding both proven separations and several open questions. Overall, the findings deepen the classification of EDM strength without AC and illuminate the model-theoretic boundaries of weak choice principles.

Abstract

We continue the study of the Erdős-Dushnik-Miller theorem (A graph with an uncountable set of vertices has either an infinite independent set or an uncountable clique) in set theory without the axiom of choice. We show that there are three inequivalent versions of this theorem and we give some results about the positions of these versions in the deductive hierarchy of weak choice principles.

Three forms of the Erdős-Dushnik-Miller Theorem

TL;DR

The paper advances the understanding of the Erdős-Dushnik-Miller theorem in the absence of the axiom of choice by distinguishing three inequivalent EDM forms: , , and . It establishes a compact trichotomy of EDM forms in the ZF weak-choice hierarchy, and uses Fraenkel-Mostowski models to separate these forms from various weak choice principles, mapping where EDM variants sit relative to PC and AC. The results include proving EDM is equivalent to AC, EDM and EDM relations in several FM models, and identifying models where EDM variants withstand or fail under symmetry constraints. The work provides a detailed landscape of how EDM interacts with PC inequalities, UT, and AC in multiple FM constructions, yielding both proven separations and several open questions. Overall, the findings deepen the classification of EDM strength without AC and illuminate the model-theoretic boundaries of weak choice principles.

Abstract

We continue the study of the Erdős-Dushnik-Miller theorem (A graph with an uncountable set of vertices has either an infinite independent set or an uncountable clique) in set theory without the axiom of choice. We show that there are three inequivalent versions of this theorem and we give some results about the positions of these versions in the deductive hierarchy of weak choice principles.

Paper Structure

This paper contains 17 sections, 33 theorems, 43 equations.

Table of Contents

  1. Introduction
  2. Definitions
  3. Weak Forms of $\mathsf{AC}$
  4. Reducing the Number of Forms of $\mathsf{EDM}$ to Three
  5. $\mathsf{EDM}$ and $\mathsf{PC}$: Positive Results
  6. Fraenkel-Mostowski Models and $\mathsf{EDM}$
  7. Terminology and Properties of Fraenkel-Mostowski models
  8. The Models $\mathcal{N}1$ and $\mathcal{N}3$
  9. The Model $\mathcal{N}2$
  10. The Model $\mathcal{N}5$
  11. The Model $\mathcal{N}_T$
  12. The Model $\mathcal{N}12(\aleph_1)$
  13. The Model $\mathcal{N}9$
  14. The Model $\mathcal{N}26$
  15. The Model $\mathcal{N}17$
  16. ...and 2 more sections

Key Result

Theorem 1.1

Any infinite graph $G = (V,E)$ not containing an independent set of size $\aleph_0$ contains a complete subgraph of size $|V|$.

Theorems & Definitions (48)

  • Theorem 1.1: $\mathsf{EDM}$
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Proposition 4.1
  • Proposition 4.2
  • Proposition 4.3
  • Proposition 4.4
  • Definition 4.5
  • Proposition 5.1
  • ...and 38 more