It is well known that from a free (non-principal) ultrafilter on $\omega$ one can define a non-measurable set of reals. The older example of a non-measurable set is the Vitali set, 
a set of representatives for the equivalence classes of the relation on the reals "the same modulo a rational number". Is it known whether you can have one without the other?
I.e., is ZF consistent with the existence of a set of representatives for the Vitali equivalence relation without having a free ultrafilter on $\omega$?


What about the other direction? I thought I had convinced myself that using an ultrafilter, you can choose representatives for the Vitali equivalence relation, but right now that does not seem clear to me anymore.



Axiom of choice: ultrafilter vs. Vitali set
Stefan, "low" cardinalities do not change by passing from $L({\mathbb R})$ to $L({\mathbb R})[{\mathcal U}]$, so the answer to the second question is that the existence of a nonprincipal ultrafilter does not imply the existence of a Vitali set.


More precisely: Assume determinacy in $L({\mathbb R})$. Then $2^\omega/E\_0$ is a successor cardinal to ${\mathfrak c}$ (This doesn't matter, all we need is that it is strictly larger. That it is a successor is a result of Richard Ketchersid and I in our forthcoming paper on $G\_0$-dichotomies, though it was long suspected. It is my understanding that it also follows from unpublished work by Foreman and Magidor).


Force with ${\mathcal P}(\omega)/Fin$ to add a Ramsey ultrafilter, so you are in the model studied by Di Prisco-Todorcevic. (The model was first studied by J.M. Henle, A.R.D. Mathias, and W.H. Woodin, in "[A barren extension](https://doi.org/10.1007/BFb0075312)", in Methods in Mathematical Logic, Lecture Notes in Mathematics 1130, Springer-Verlag, 1985, pages 195-207, where they show for example that no new sets of ordinals are added in this extension.)


In their forthcoming paper on "[Borel cardinals and Ramsey ultrafilters](http://www.users.miamioh.edu/larsonpb/ult_card_512.pdf)" by Ketchersid, Larson, and Zapletal, the question of how the (non-well-ordered) cardinality structure changes by going to this model is studied. I believe there are still many questions left, but one of the problems they have settled is in showing that $2^\omega/E\_0$ is still strictly larger than ${\mathfrak c}$. This means we cannot pick representatives of the Vitali classes, of course (if $\sim$ is the Vitali equivalence relation, then ${\mathbb R}/\sim$ and $2^\omega/E\_0$ are ``Borel isomorphic''), or else we would have that $2^\omega/E\_0$ and $2^\omega$ have the same size by Schroeder-Bernstein.


