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For your second definition of Vitali set, I have a weak partial answer. Namely the existence of a Bernstein set does not imply the existence of a $T$-Vitali set. The answer can be found in logic blog maintained by Andre Nies:
http://dl.dropbox.com/u/370127/Blog/Blog2012.pdf

Note that a Turing degree does not need to be a an addition group.

I don't know whether the existence of a Vitali set implies the existence of a Bernstein set. But it is not difficult to see, under $ZF+DC$, that there is a Vitali set (if it exists) which contains a perfect subset.

For you first definition of Vitali set, I have no idea.

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For your second definition of Vitali set, I have a weak partial answer. Namely the existence of a Bernstein set does not imply the existence of a Vitali $T$-Vitali set. The answer can be found in logic blog maintained by Andre Nies:
http://dl.dropbox.com/u/370127/Blog/Blog2012.pdf

Note that a Turing degree does not need to be a addition group

I don't know whether the existence of a Vitali set implies the existence of a Bernstein set. But it is not difficult to see, under $ZF+DC$, that there is a Vitali set (if it exists) which contains a perfect subset.

For you first definition of Vitali set, I have no idea.

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For your second definition of Vitali set, I have a partial answer. Namely the existence of a Bernstein set does not imply the existence of a Vitali set. The answer can be found in logic blog maintained by Andre Nies:
http://dl.dropbox.com/u/370127/Blog/Blog2012.pdf

I don't know whether the existence of a Vitali set implies the existence of a Bernstein set. But it is not difficult to see, under $ZF+DC$, that there is a Vitali set (if it exists) which contains a perfect subset.

For you first definition of Vitali set, I have no idea.