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 $\omega\_2\cap cof(\omega\_1)$?



Saturation of non-stationary ideal on $\omega_2$?
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 But it was known by early unpublished work of Woodin that $\mathrm{NS}\_{\omega\_2} \restriction S$ can be saturated, where $S$ is a stationary-costationary subset of $\mathrm{cof}(\omega\_1)$.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.](https://arxiv.org/abs/1901.02821)


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.


