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!



Strong partition property + DC + existence of non-principal ultrafilter on $\omega$
The key reference for this is



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> [MR0799042 (87d:03141)](https://mathscinet.ams.org/mathscinet-getitem?mr=799042). Henle, J. M.; Mathias, A. R. D.; Woodin, W. Hugh. *[A barren extension](https://link.springer.com/chapter/10.1007%2FBFb0075312)*. In **Methods in mathematical logic (Caracas, 1983)**, C. A. Di Prisco, editor, 195–207, Lecture Notes in Math., 1130, Springer, Berlin, 1985.
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There, Henle, Mathias, and Woodin start with $L(\mathbb R)$ under the assumption of determinacy (and [$\mathsf{DC}$](https://mathscinet.ams.org/mathscinet-getitem?mr=736611)), 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).


