Are there any non-amenable group $G$ with the property:

There exists $C<1$ such that for every finite set $S\subset G$ there exists a set $F\subseteq S$ such that $|F|\geq C |S|$ and $F$ generates an amenable group.

UPDATE: Here is some motivation for the question: http://mathoverflow.net/questions/55075/amenability-of-groups-iii

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# Amenability of groups II

Are there any non-amenable group $G$ with the property:

There exists $C<1$ such that for every finite set $S\subset G$ there exists a set $F\subseteq S$ such that $|F|\geq C |S|$ and $F$ generates an amenable group.