Square-bracket operations clubs

TL;DR

This paper investigates the Ramsey club property for square-bracket operations on $\omega_1$, showing its independence from ZFC and analyzing its status under various set-theoretic axioms.

Contribution

It extends the known results about the Ramsey club property from Aronszajn trees to other classes of square-bracket operations and establishes its independence from ZFC.

Findings

Proper Forcing Axiom implies all such operations have the property.

The property is independent of ZFC, consistent with both its presence and absence.

The status under Martin's Axiom and CH is also analyzed.

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 $C\subseteq\omega_{1}$, there is an uncountable subset $W$ $\subseteq \omega_{1}$ such that $\left[ \alpha\beta\right] \in C$ for every $\alpha,\beta\in W.$ \ 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.