Club Chang's Conjecture

TL;DR

This paper explores a stronger form of Chang's Conjecture, called Club-CC, which involves the existence of closed unbounded sets within certain models, and demonstrates its consistency under stronger large cardinal assumptions.

Contribution

It introduces the principle Club-CC, proves its consistency with stronger large cardinal assumptions, and shows it implies the failure of certain weak square principles.

Findings

Club-CC is consistent under stronger large cardinal assumptions.

Club-CC implies the failure of certain weak square principles.

It extends the classical Chang's Conjecture with additional structural properties.

Abstract

Chang's Conjecture (CC) asserts that for every $F:[\omega_2]^{<\omega} \to \omega_2$, there exists an $X$ that is closed under $F$ such that $|X|=\omega_1$ and $|X \cap \omega_1| =\omega$. By classic results of Silver and Donder, CC is equiconsistent with an $\omega_1$-Erdos cardinal. Using stronger large cardinal assumptions (between $o(\kappa) = \kappa^+$ and $o(\kappa) = \kappa^{++}$), we prove that it is consistent to also require that $X$ contains a closed unbounded set of ordinals in $\text{sup}(X \cap \omega_2)$. We denote this stronger principle \textbf{Club-CC}, and also show that, unlike CC, Club-CC implies failure of certain weak square principles.