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 $\omega_2\cap cof(\omega_1)$?
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2$\begingroup$ In a recent preprint ( arxiv.org/abs/1901.02821 ) Monroe Eskew proved that it is consistent relative to a huge cardinal that local saturation holds simultaneously at all successor cardinals. $\endgroup$– Yair HayutCommented Dec 3, 2019 at 18:26
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2$\begingroup$ the full set of critical cofinalities case seems not covered there $\endgroup$– OttoCommented Dec 3, 2019 at 23:03
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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 But it was known by early unpublished work of Woodin that $\mathrm{NS}_{\omega_2} \restriction S$ can be saturated, where $S$ is a stationary-costationary subset of $\mathrm{cof}(\omega_1)$.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.
answered Dec 4, 2019 at 13:07
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$\begingroup$ Is it known if it can be pre-saturated? $\endgroup$ Commented Dec 4, 2019 at 18:40
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1$\begingroup$ @Rahman.M I think this is also open. $\endgroup$ Commented Dec 5, 2019 at 22:45