Local saturation and square everywhere

Abstract

We show that it is consistent relative to a huge cardinal that for all infinite cardinals $\kappa$, ${\square }_{\kappa }$ holds and there is a stationary $S\subseteq {\kappa }^{+}$ such that ${\mathrm{NS}}_{{\kappa }^{+}}\upharpoonright S$ is ${\kappa }^{++}$-saturated.