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| Main Authors: | , |
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| Format: | Preprint |
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2026
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| Online Access: | https://arxiv.org/abs/2602.14373 |
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| _version_ | 1866911449842974720 |
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| author | Guzman, Osvaldo Todorcevic, Stevo |
| author_facet | Guzman, Osvaldo Todorcevic, Stevo |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_14373 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Square-bracket operations clubs Guzman, Osvaldo Todorcevic, Stevo Logic 03E05, 3E50, 03E57, 03E35 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. |
| title | Square-bracket operations clubs |
| topic | Logic 03E05, 3E50, 03E57, 03E35 |
| url | https://arxiv.org/abs/2602.14373 |