Chang's conjecture and weak square
In this well-written, self-contained paper, the author discusses the influence of variations of Chang's conjecture on weak square principles. He first showed that for all infinite cardinals \(\mu<\nu\), if \((\nu^+, \nu)\twoheadrightarrow(\mu^+, \mu)\) holds, then there is a \({<}\nu^+\)-Baire forcing extension in which both \((\nu^+, \nu)\twoheadrightarrow(\mu^+, \mu)\) and \(\square(\nu^+)\) hold. Note that \textit{S. Todorčević} [NATO ASI Ser., Ser. C, Math. Phys. Sci. 411, 385--398 (1993; Zbl 0844.03027)] proved that \((\nu^+, \nu)\twoheadrightarrow(\mu^+, \mu)\) implies the failure of \(\square_\nu\). He also discusses the relationship between \((\nu^+, \nu)\twoheadrightarrow(\mu^+, \mu)\) and the weak square of the form \(\square_{\nu, \rho}\) when \(\nu\) is regular. The case when \(\nu\) is singular was already investigated by \textit{M. Foreman} [DIMACS, Ser. Discrete Math. Theor. Comput. Sci. 58, 73--94 (2002; Zbl 1013.03049)]. When \(\mu=\omega\), he proved that if \(\nu\) is an uncountable regular cardinal and there is a measurable cardinal above \(\nu\), then there is a \({<}\nu\)-Baire forcing extension in which both \((\nu^+, \nu)\twoheadrightarrow(\mu^+, \mu)\) and \(\square_{\omega_1, 2}\) hold. He built such a model by first using Levy collapse to make the measurable cardinal to \(\nu^+\) and then forcing \(\square_{\nu, 2}\). When \(\mu\) is uncountable, he obtained the following result: for all uncountable cardinals \(\mu<\nu\), \(\square_{\nu, \mu}\) fails if \((\nu^+, \nu)\twoheadrightarrow(\mu^+, \mu)\) holds, and either (I) \(\nu^{{<}\nu}=\nu\) or (II) \(|\{\rho\in\mathrm{Reg} \mid \mu<\rho\leq\nu\}|<|\{\rho\in\mathrm{Reg} \mid \omega\leq\rho\leq\mu\}|<\aleph_1\). Here, \(\mathrm{Reg}\) denotes the class of all regular cardinal. For example, it follows that \((\omega_3, \omega_2)\twoheadrightarrow(\omega_2, \omega_1)\) implies the failure of \(\square_{\omega_2, \omega_1}\).
scientific article; zbMATH DE number 6136595