It is a well known result of Kleinberg that the axiom of determinacy implies that the are Jónsson cardinals for all . This follows from a careful analysis of partition properties and the computation of ultrapowers of associated measures. See also here for extensions of this result, references, and some additional context.

Using his theory of descriptions of measures, Steve Jackson proved (in unpublished work) that, assuming determinacy, every cardinal below is in fact Rowbottom. See for example these slides: 1, 2. Woodin mentioned after attending a talk on this result that the HOD analysis shows that every cardinal is Jónsson below .

During the Second Conference on the Core Model Induction and HOD Mice at Münster, Jackson, Ketchersid, and Schlutzenberg reconstructed what we believe is Woodin’s argument.

I have posted a short note with the proof on my papers page. The note is hastily written, and I would appreciate any

additions/comments/corrections/suggestions

you ﬁnd appropriate. I posted a version of the note in August last year, and Grigor Sargsyan soon emailed me some comments, that I have incorporated in the current version.

[I mentioned this on Twitter last year, but apparently not here. These are the relevant links (while discussing the Münster meeting): 1, 2, 3, 4, 5.]