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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 IIAre 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.
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