TITLE: Generalising Vitali Sets to uncountable dense subgroup selectors...
QUESTION [5 upvotes]: Does there exist an uncountable dense subgroup, $\Gamma$, of the additive group $\mathbb{R}$, such that every selector of the partition of $\mathbb{R}$ canonically associated with the equivalence relation $x \in \mathbb{R}$ & $y \in \mathbb{R}$ & $x − y \in \Gamma$ is nonmeasurable?
This question is related to a previous question of mine which has not been answered yet.

REPLY [8 votes]: I edited my former answer in light of a recent e-mail exchange with Sławomir Solecki.
There is a substantial literature on Vitali-like selectors. A major paper in the area is the following, available here.
Cichoń, J.; Kharazishvili, A.; Węglorz, B. On sets of Vitali's type. Proc. Amer. Math. Soc. 118 (1993), no. 4, 1243–1250.
There are many interesting results in the above paper, including the fact, observed by Joel Hamkins in his answer, that $MA+\lnot CH$ implies the existence of uncountable subgroups $G$ of $\Bbb{R}$ with no measurable $G$-selectors; where a $G$-selector is a choice function that selects one element from each $G$-coset.

Moreover, Theorem 6 of the above paper shows that under $MA+\lnot CH$ there is even a subgroup $G$ of $\Bbb{R}$ of power $2^{\aleph_0}$ that has no measurable $G$-selector.

This is to be contrasted with Theorem 2 of the paper, which asserts every analytic subgroup $G$ of reals has a measurable $G$-selector.

According to Solecki (private communication) it is apparently unknown if $ZFC$ can prove that there is an uncountable subgroup of the reals all of whose selectors are Lebesgue non-measurable.

However, if one allows extensions of measures, then there is a fundamental distinction to be made (note: the theorem below was first stated and proved with $CH$; but $CH$ was subsequently removed in the paper cited below).

Theorem (Solecki). The following are equivalent for every subgroup  $G$ of reals.
(a) $G$ is countable and dense.
(b) Every $G$-selector is non-measurable with respect to any translation invariant extension of the Lebesgue measure.

Solecki's result above is a special case of Cor. 2.2. of the following paper of his:
Translation invariant ideals, Israel J. Math. 135 (2003), 93--110