The paper [Generic $I_0$ at $\aleph_\omega$](http://arxiv.org/pdf/1609.07970.pdf) by Vincenzo Dimonte might be of interest to you. It introduces the notion of being generic $I_0$ at $\aleph_\omega$ (Def. 3.1 of the paper), and proves several consequences of it. In particular it is shown that if eneric $I_0$ holds at $\aleph_\omega$, then $\aleph_\omega$ is Jonsson.

Of course it is open if generic $I_0$ at $\aleph_\omega$ can be consistent!.