Square principles with tail-end agreement

Abstract

This paper investigates the principles ${\square^{{{\rm ta}}}_{\lambda,\delta}}$ , weakenings of ${\square_\lambda}$ which allow ${\delta}$ many clubs at each level but require them to agree on a tail-end. First, we prove that ${\square^{{\rm {ta}}}_{\lambda,< \omega}}$ implies ${\square_\lambda}$. Then, by forcing from a model with a measurable cardinal, we show that ${\square_{\lambda,2}}$ does not imply ${\square^{{\rm{ta}}}_{\lambda,\delta}}$ for regular ${\lambda}$ , and ${\square^{{\rm{ta}}}_{\delta^+,\delta}}$ does not imply ${\square_{\delta^+,< \delta}}$ . With a supercompact cardinal the former result can be extended to singular λ, and the latter can be improved to show that ${\square^{{\rm {ta}}}_{\lambda,\delta}}$ does not imply ${\square_{\lambda,< \delta}}$ for ${\delta < \lambda}$.