It was mentioned after Theorem 30.27 in Kanamori's Higher Infinite that Woodin constructed a model of $DC$ + there exists unboundedly many many $\kappa<\Theta$ such that $\kappa \to (\kappa)^\kappa_{\alpha} \ \forall \alpha<\kappa$ and there exists a non-principal ultrafilter on $\omega$. In particular, this model is not a model of $AD$.
What is the reference of this? Or even better for those who know the proof, what does the proof look like? Thanks!
The key reference for this is
MR0799042 (87d:03141). Henle, J. M.; Mathias, A. R. D.; Woodin, W. Hugh. A barren extension. In Methods in mathematical logic (Caracas, 1983), C. A. Di Prisco, editor, 195–207, Lecture Notes in Math., 1130, Springer, Berlin, 1985.
There, Henle, Mathias, and Woodin start with $L(\mathbb R)$ under the assumption of determinacy (and $\mathsf{DC}$), and force with $\mathcal P(\omega)/\mathrm{Fin}$; they refer to the resulting model as "the Hausdorff extension".
They use $\mathsf{ZF}+\omega\to(\omega)^\omega$ to prove that the Hausdorff extension is barren, meaning that every map from an ordinal into the ground model was already in the ground model. They also show (under $\mathsf{AD}+V=L(\mathbb R)$) that all strong partition cardinals below $\Theta$ are preserved in the extension. Easily, the extension also preserves $\mathsf{DC}$.
On the other hand, the forcing adds a Ramsey ultrafilter on $\omega$ (in particular, unboundedly many strong partition cardinals below $\Theta$ is not enough to ensure all sets all reals are Lebesgue measurable).
