I've recently encountered the notion of a universally Baire set, and I've tried to look at the paper by Feng, Magidor and Woodin where this notion is studied. There are several points that confuse me.

(1) First of all, what is the extend to which we know that various subsets of the reals have tree representations, i.e., are $\kappa$-Souslin for some cardinal $\kappa$? Is it correct that under AD, all projective set have a tree representation? What about in Solovay's model? Do all subsets of the reals there have tree representations? (It is not at all clear to me that being definable from a sequence of ordinals has anything to do with having a tree representation, but perhaps I am missing something obvious.)

(2) These points aside, what I am really interested in is trying to understand the relationship between the notion of a universally Baire set and the paper by Shelah called "Can you take Solovay's inaccessible away?" (Isr. J. Math, 1984). It at some point seemed to me that what maybe really is going on there (in the section about a model of ZF where all sets have the Baire property) is that Shelah is proving that there is a model of ZF in which all sets are $\omega$ universally Baire, or something close to this fact.

(Here, when I say $\omega$ universally Baire, I mean a subset $A$ of Baire space where there are trees $T$ and $T^*$ such that $p[T]=A$, $p[T^*]=A^c$, and in any forcing extension by a countable poset (Cohen forcing) we have $p[T]\cup p[T^{*}]=\omega^\omega$.)

So, basically my question is: Is there a model of ZF where all subsets of the reals are $\omega$ universally Baire? And if yes, does this require a large cardinal hypothesis?

I hope someone can help, as I am completely new to much of this. Even just some references where some basic things are more clearly explained would help a lot. Thanks, R.A.D.

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to encase them. – Asaf Karagila Apr 6 at 15:58