Consistency strength of saturated ideals on $[\lambda]^{<\kappa}$

In the Handbook of Set Theory, Foreman notes that in models constructed by Magidor from a huge cardinal, there exists a normal ideal on $[\omega_2]^{<\omega_1}$ that is $\omega_3$-saturated. Other results in this article imply that this is the minimal possible saturation, and such an ideal carries strong properties: Any generic ultrapower is well-founded and contains the same reals as the forcing extension.

Is there anything known about lower bounds on the consistency strength of normal, $\omega_{n+1}$-saturated ideals on $[\omega_n]^{<\omega_1}$ for $n>1$? For $n = 1$, this is known to be equiconsistent with a Woodin cardinal, by results of Shelah and recent work of Jensen and Steel. (Thanks, Andres!)

• You can remove the "almost", see here. – Andrés E. Caicedo Aug 17 '13 at 3:13
• @AndrésCaicedo, Can you explain the significance of the Jensen-Steel result? It seems Schindler already had the result here: mittag-leffler.se/sites/default/files/IML-0910f-05.pdf – Monroe Eskew Mar 23 '16 at 18:53
• Hi, Monroe. Ralf is assuming more than the saturation of the ideal (or rather, he is assuming less on the ideal, but in conjunction with another hypothesis). Similarly, John's original result assumes the saturation of the ideal, and an extra hypothesis. In both cases, the additional assumption gives us enough partial measures that we can easily build "local" versions of $K$. The point of the Jensen-Steel result is that they carry out the construction of $K$ without the need for the additional assumptions. – Andrés E. Caicedo Mar 24 '16 at 1:19
• @AndrésCaicedo, are you sure? This is not explicitly stated in Theorem 0.6 in the link. – Monroe Eskew Mar 24 '16 at 11:22
• Ah, I see now what you were asking. The link you gave is to an early version. The published reference is MR2963017. Claverie, Benjamin(D-MUNS-ML); Schindler, Ralf(D-MUNS-ML). Woodin's axiom $(∗)$, bounded forcing axioms, and precipitous ideals on $\omega_1$. J. Symbolic Logic 77 (2012), no. 2, 475–498. There they explain how to prove the equiconsistency of a Woodin cardinals with a strong ideal on $\omega_1$. They explicitly make use of the Jensen-Steel approach (which, in turn, uses the technique of stacking mice) and some new ideas. – Andrés E. Caicedo Mar 24 '16 at 15:25