Suppose that $\kappa$ is a regular cardinal and let $NS$ be the ideal of its nonstationary subsets. One can consider the Boolean algebra $P(\kappa) /NS$ and say that (if $\lambda$ is another cardinal) $NS$ is $\lambda$ saturated iff there are no antichains in $P(\kappa) / NS$ of length $\lambda$. It is an elegant result of Gitik and Shelah that $NS$ cannot be $\kappa^+$ saturated for every regular $\kappa > \aleph\_1$, on the other hand Foreman Magidor and Shelah could show that assuming a supercompact cardinal it is consistent that $NS$ of $\omega\_1$ is $\aleph\_2$ saturated. These results are all well known and one can find them for example in Jech's book. However it is stated there that Shelah eventually found that even a Woodin cardinal suffices to obtain the consistency of the statement "$NS$ on $\omega\_1$ is $\aleph\_2$ saturated". 


Do you know where I can find a proof of this result? 


Thank you



A result of Shelah about the nonstationary ideal
Try also Chapter XVI of "Proper and Improper Forcing" (entitled "Large ideals on $\aleph\_1$ from smaller cardinals"). It's hard to tell exactly what's in there, but he does say in the chapter he will "keep old promises from 84-85 mentioned in [Sh:253]", where [Sh:253] is the paper Michael mentions, and he does claim to be replacing certain hypotheses used earlier by the assumption "$\lambda$ is a Woodin cardinal".


