This question appeared in my answer to [this](https://mathoverflow.net/questions/54838/amenability-of-groups-ii)  question, but it seems to be interesting in itself. Let $G$ be an infinite finitely generated group, $\epsilon\gt 0$. Is there a finite subset $S\subset G$ such that every subset of $S$ with at least $\epsilon|S|$ elements generates $G$? If the answer is "yes", it should have a trivial proof by Gromov's thesis (every property of all finitely generated groups is either false or trivial). 


 **Update.**  In view of Stephen's answer and Kevin's comment below, perhaps a more correct question is this: 


* Is it true that if we represent an infinite group $G$ as a union of a finite number of subsets, then one of these subsets generates a finite index subgroup of $G$?


Compare with Adreas Thom's [question](https://mathoverflow.net/questions/54921/generation-of-finite-index-subgroups).



A non-trivial property of all groups
This is false for the infinite dihedral group $\langle a,b\mid b^2=1, ba=a^{-1}b\rangle$. No set $S$ works for $\epsilon\le1/3$, because there is always a subset with $\lceil{\epsilon|S|}\rceil$ elements that lies entirely in $\{a^n\mid n\in\mathbb{Z}\}$, $\{a^{2n}b\mid n\in\mathbb{Z}\}$ or $\{a^{2n+1}b\mid n\in\mathbb{Z}\}$, and none of these three sets generates the whole group.


