Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Walks on uncountable ordinals and non-structure theorems for higher Aronszajn lines

Tanmay Inamdar, Assaf Rinot

Abstract

It is proved that if there is an $\aleph_2$-Aronszajn line, then there is one that does not contain an $\aleph_2$-Countryman line. This solves a problem of Moore and stands in a sharp contrast with his Basis Theorem for linear orders of size $\aleph_1$. The proof combines walks on ordinals, club guessing, strong colourings of three different types, and a bit of finite combinatorics. This and further non-structure theorems for Aronszajn lines and trees are established for successors of regulars, successors of singulars, as well as inaccessibles.

Walks on uncountable ordinals and non-structure theorems for higher Aronszajn lines

Abstract

It is proved that if there is an -Aronszajn line, then there is one that does not contain an -Countryman line. This solves a problem of Moore and stands in a sharp contrast with his Basis Theorem for linear orders of size . The proof combines walks on ordinals, club guessing, strong colourings of three different types, and a bit of finite combinatorics. This and further non-structure theorems for Aronszajn lines and trees are established for successors of regulars, successors of singulars, as well as inaccessibles.

Paper Structure

This paper contains 20 sections, 48 theorems, 109 equations.

Table of Contents

  1. Introduction
  2. Classifying uncountable structures
  3. Non-structure theory for higher Aronszajn lines
  4. Upcoming results
  5. Organization of this paper
  6. Notation and conventions
  7. Trees and lines
  8. Colourings
  9. A question of Erdős and Hajnal
  10. Projection maps
  11. $C$-sequences, walks on ordinals and club-guessing
  12. Walks on ordinals
  13. Club-guessing
  14. T(rho0)
  15. T(rho1)
  16. ...and 5 more sections

Key Result

Theorem A

Suppose that $\kappa=\mu^+$ for a regular uncountable cardinal $\mu$ that is non-ineffable. Then all of the following are equivalent:

Theorems & Definitions (146)

  • Definition 1.2
  • Theorem A
  • Theorem B
  • Theorem C
  • Theorem D
  • Definition 2.1: Lücke, Lucke
  • Definition 2.2: Todorčević, TodActa
  • Remark 2.3
  • Definition 2.4
  • Lemma 2.5: folklore
  • ...and 136 more