Diagonal reflections on squares

Abstract

The effects of (bounded versions of) the forcing axioms $\mathsf {SCFA}$, $\mathsf {PFA}$ and $\mathsf {MM}$ on the failure of weak threaded square principles of the form $\square (\lambda ,\kappa )$ are analyzed. To this end, a diagonal reflection principle, $\mathsf {DSR}{\left( {<}\kappa ,S\right) }$ is introduced. It is shown that $\mathsf {SCFA}$ implies $\mathsf {DSR}{\left( \omega _1,S^\lambda _\omega \right) }$, for all regular $\lambda \ge \omega _2$, and that $\mathsf {DSR}{\left( \omega _1,S^\lambda _\omega \right) }$ implies the failure of $\square (\lambda ,\omega _1)$ if $\lambda >\omega _2$, and it implies the failure of $\square (\lambda ,\omega )$ if $\lambda =\omega _2$. It is also shown that this result is sharp. It is noted that $\mathsf {MM}$/$\mathsf {PFA}$ imply the failure of $\square (\lambda ,\omega _1)$, for every regular $\lambda >\omega _1$, and that this result is sharp as well.