Club Chang's Conjecture

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.

arXiv commentTheorem 12, and the proof of Claim 13, are not correct (thanks to Omer Ben-Neria for pointing this out). The notion in Definition 7 is inconsistent with ZFC