Question: Does Con($ZF$) imply Con($ZF$ + $DC$ + "there is no paradoxical Banach-Tarski decomposition of the unit ball")?
Here Con($X$) is the consistency of $X$; $DC$ is dependent choice.
Motivation for the Question: Since the "paradoxical" sets in the Banach-Tarski theorem have to be non-measurable, Solovay's model of "every subset of reals is measurable" shows:
Theorem. Con($ZF$+ Inacc) implies Con($ZF$ + $DC$ + "there is no paradoxical Banach-Tarski decomposition of the unit ball").
In the above Inacc is the statement "there is an inaccessible cardinal".
Note that Shelah's model [Israel Journal of Math, 1984], constructed only from Con($ZF$), in which $DC$ holds and all sets of reals have the Baire property, is of no help in answering the above question since Dougherty and Foreman [ J. Amer. Math. Soc., 1994] have shown that there are paradoxical decompositions of the unit ball using pieces which have the property of Baire. I do not know how much choice is needed in their construction.
This question was posed a while ago on an FOM-posting of mine, but remained unanswered.