TITLE: Saturation of non-stationary ideal on $\omega_2$?
QUESTION [6 upvotes]: It is known that $NS_{\omega_2}$ cannot be saturated (namely there cannot be $\aleph_3$ many stationary subsets of $\omega_2$ any two of which have non-stationary intersection). However, it may be the case when it is restricted to a stationary subset. It is also known that the stationary subset cannot be $\omega_2\cap cof(\omega)$. Can $NS_{\omega_2}\restriction cof(\omega_1)$ be saturated? or on some stationary subset of $\omega_2\cap cof(\omega_1)$?

REPLY [3 votes]: I believe it is still open whether $\mathrm{NS}_{\omega_2} \restriction \mathrm{cof}(\omega_1)$ can be saturated.  But it was known by early unpublished work of Woodin that $\mathrm{NS}_{\omega_2} \restriction S$ can be saturated, where $S$ is a stationary-costationary subset of $\mathrm{cof}(\omega_1)$.  The complement provides a "safe space" to allow the iteration to work.  Details are given by Foreman-Komjath (MR: 2151585) and by me.
I have heard rumors that recent work related to higher analogues of forcing axioms might be used to tackle the problem.  This approach would surely give a model with not-GCH, so we'd have an obvious next open question to tackle.