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Erdős-Rado Classes

Will Boney

Abstract

We amalgamate two generalizations of Ramsey's Theorem--Ramsey classes and the Erdős-Rado Theorem--into the notion of a combinatorial Erdős-Rado class. These classes are closely related to Erdős-Rado classes, which are those from which we can build generalized indiscernibles and blueprints in nonelementary classes, especially Abstract Elementary Classes. We give several examples and some applications.

Erdős-Rado Classes

Abstract

We amalgamate two generalizations of Ramsey's Theorem--Ramsey classes and the Erdős-Rado Theorem--into the notion of a combinatorial Erdős-Rado class. These classes are closely related to Erdős-Rado classes, which are those from which we can build generalized indiscernibles and blueprints in nonelementary classes, especially Abstract Elementary Classes. We give several examples and some applications.

Paper Structure

This paper contains 20 sections, 23 theorems, 67 equations.

Table of Contents

  1. Introduction
  2. Outline
  3. Conventions
  4. Preliminaries
  5. Classes of structures and types
  6. Generalized indiscernibles and blueprints
  7. Our examples
  8. Structural Partition Relations and Combinatorial Erdős-Rado Classes
  9. Examples and some counter-examples
  10. Erdős-Rado Classes and the Generalized Morley's Omitting Types Theorem
  11. Further results
  12. Reversing Generalized Morley's Omitting Types Theorem
  13. A category theoretic interpretation of blueprints
  14. Generalized Shelah's Omitting Types Theorem
  15. End-approximations
  16. ...and 5 more sections

Key Result

Proposition 2.3

Fix a language $\tau$. Then

Theorems & Definitions (67)

  • Definition 2.1
  • Definition 2.2
  • Proposition 2.3
  • Definition 2.4
  • Example 2.5
  • Definition 2.6
  • Example 2.7: Linear orders
  • Example 2.8: $\chi$ disjoint linear orders
  • Example 2.9: $\chi$-colored linear orders
  • Example 2.10: Trees of height $n<\omega$
  • ...and 57 more