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Square-bracket operations clubs

Osvaldo Guzman, Stevo Todorcevic

TL;DR

This work enunciates and investigates the Ramsey club property for three square-bracket operations on $\omega_1$, showing that ZFC alone cannot decide whether all such operations possess or lack the property. It extends PFA’s known Ramsey-club result from Aronszajn-tree–induced operations to the other two operation types, and demonstrates independent consistency results relative to MA and CH using Baumgartner’s club forcing and Asperó’s product. The paper constructs clubs that are Ramsey for chosen operations and, conversely, forces models where the generic club destroys Ramsey-ness for all square-bracket operations, ultimately proving that both all-and-none Ramsey-ness statements are consistent with ZFC (and with or without MA/CH in various configurations). It further analyzes how PFA, MA, CH, and related principles interact with these properties across the different operation classes, culminating in a set of open questions about deeper interactions and potential extensions.

Abstract

This paper continues the investigation of the three square-bracket operations $[\cdot\cdot]$ from chapter 5 of \cite{Walks}. \ We say that a square-bracket operation $[\cdot\cdot]$ has the \emph{Ramsey club property} if for every club $C\subseteqω_{1}$, there is an uncountable subset $W$ $\subseteq ω_{1}$ such that $\left[ αβ\right] \in C$ for every $α,β\in W.$ \ The second author proved that the Proper Forcing Axiom\textsf{ }implies that all the square-bracket operations induced by Aronszajn trees have this property. We extend this result to the other two classes. We conclude that each of the statements \textquotedblleft all square-bracket operations have the Ramsey club property\textquotedblright\ and \textquotedblleft No square-bracket operation has the Ramsey club property\textquotedblright\ are consistent with \textsf{ZFC. }In other words, \textsf{ZFC }is unable to decide the status of the Ramsey club property for any square-bracket operation. Furthermore, we analyze the status of the Ramsey club property for square-bracket operations under Martin's Axiom and the Continuum Hypothesis.

Square-bracket operations clubs

TL;DR

This work enunciates and investigates the Ramsey club property for three square-bracket operations on , showing that ZFC alone cannot decide whether all such operations possess or lack the property. It extends PFA’s known Ramsey-club result from Aronszajn-tree–induced operations to the other two operation types, and demonstrates independent consistency results relative to MA and CH using Baumgartner’s club forcing and Asperó’s product. The paper constructs clubs that are Ramsey for chosen operations and, conversely, forces models where the generic club destroys Ramsey-ness for all square-bracket operations, ultimately proving that both all-and-none Ramsey-ness statements are consistent with ZFC (and with or without MA/CH in various configurations). It further analyzes how PFA, MA, CH, and related principles interact with these properties across the different operation classes, culminating in a set of open questions about deeper interactions and potential extensions.

Abstract

This paper continues the investigation of the three square-bracket operations from chapter 5 of \cite{Walks}. \ We say that a square-bracket operation has the \emph{Ramsey club property} if for every club , there is an uncountable subset such that for every \ The second author proved that the Proper Forcing Axiom\textsf{ }implies that all the square-bracket operations induced by Aronszajn trees have this property. We extend this result to the other two classes. We conclude that each of the statements \textquotedblleft all square-bracket operations have the Ramsey club property\textquotedblright\ and \textquotedblleft No square-bracket operation has the Ramsey club property\textquotedblright\ are consistent with \textsf{ZFC. }In other words, \textsf{ZFC }is unable to decide the status of the Ramsey club property for any square-bracket operation. Furthermore, we analyze the status of the Ramsey club property for square-bracket operations under Martin's Axiom and the Continuum Hypothesis.
Paper Structure (15 sections, 55 theorems)

This paper contains 15 sections, 55 theorems.

Table of Contents

  1. Introduction
  2. Notation
  3. Preliminaries on countable elementary submodels
  4. Preliminaries on Aronszajn trees
  5. Preliminaries on forcing
  6. Preliminaries on walks on ordinals
  7. Preliminaries on Baumgartner's club forcing
  8. The square-bracket operations
  9. Finding Ramsey clubs
  10. Destroying square-bracket operations
  11. Taking care of all square-bracket operations
  12. Square-bracket operations and CH
  13. PFA and square-bracket operations induced by a sequence of reals
  14. PFA and square-bracket operations induced by C-sequences
  15. Open questions

Key Result

Theorem 1

There is $c:\left[ \omega _{1}\right] ^{2}\longrightarrow\omega_{1}$ such that for every $A\in\left[ \omega_{1}\right] ^{\omega_{1}}$ and $\gamma\in\omega_{1}$, there are $\alpha,\beta\in A$ such that $c\left( \alpha,\beta\right) =\gamma.$

Theorems & Definitions (79)

  • Theorem 1: T. Walks, PartitioningPairs
  • Theorem 2: T. Walks
  • Definition 3
  • Theorem 4: T.
  • Corollary 5
  • Definition 6
  • Theorem 7: T. Walks
  • Proposition 8
  • Lemma 9
  • Definition 10
  • ...and 69 more