An ideal on $\omega\_1$ is *strong* if it is precipitous and the associated generic elementary embedding always maps $\omega\_1$ to $\omega\_2$. This definition is from Baumgartner and Taylor, *Saturation Properties of Ideals in Generic Extensions II*
(available online at <http://www.jstor.org/stable/1998900>.) Every pre-saturated ideal on $\omega\_1$ is strong (this was well-known even before the terminology was introduced, I think) and in this paper the authors ask whether the converse is true.


Does anyone know the status of this question: "is every strong ideal on $\omega\_1$ pre-saturated"?



Strong ideals that are not pre-saturated
It is consistent that the nonstationary ideal on $\omega\_1$ is strong but not pre-saturated. Baumgartner and Taylor proved in the aforementioned paper that strong ideals are preserved by c.c.c. forcing and asked whether the same is true for pre-saturated ideals. The answer to this question is negative, implying a negative answer to the question I posted above. Apparently this was first proved by Veličković in the paper *Forcing axioms and stationary sets* (which I cannot seem to access online) from ZFC + SPFA. Another example of a c.c.c. forcing that destroys pre-saturation may be found in a more recent paper by Larson and Yorioka, *Another c.c.c. forcing that destroys presaturation*, assuming the consistency of ZF + AD.


I don't know if a negative answer can be forced from only a Woodin cardinal (which is equiconsistent with the existence of a presaturated ideal and also with the existence of a strong ideal.)


