Question

For my research I've been trying to locate Solovay's proof that AD+ implies that all sets are Ramsey (or completely Ramsey, depending on the terminology). I know that it uses Mathias forcing but I can't locate the paper. Does anyone know where I could find it or another paper that reproduces his proof using that technique?

Answer 1

About all you need is in Adrian's paper "Happy families". It is not spelled out as following from ${\sf AD}^+$, of course, since the concept did not exist yet.

Anyway, if you are familiar with Solovay's arguments in his paper on all sets of reals being measurable (or as discussed in the Feng-Magidor-Woodin paper on universally Baire sets), then all you want to read is section 2 (up to 2.2.3) of my paper with Richard Ketchersid, "A trichotomy theorem in natural models of ${\sf AD}^+$", in Set Theory and Its Applications, Contemporary Mathematics, vol. 533, Amer. Math. Soc., Providence, RI, 2011, pp. 227-258, MR2777751, and also available at my papers page.

Very briefly, what we use of ${\sf AD}^+$ is little more than all sets of reals having $\infty$-Borel codes. If $S$ is an $\infty$-Borel code for a set of reals $A$, then $L[S]$ is a model of choice, so (from determinacy), its reals are countable and in fact $\omega_1^V$ is inaccessible in $L[S]$. One can now run the argument that sets of reals are Ramsey in the Solovay model, noting that there are (in $V$) generics over $L[S]$ for Mathias's forcing, and the proof is concluded from the defining property of $\infty$-Borel sets.

Since the proof uses the $\infty$-Borel machinery so explicitly, it is still open whether ${\sf AD}$ suffices for this result.

Answer 2

So, what is the reference? I'm trying to write the bibliography for my dissertation and am desperately searching for the exact reference.