# Amenability of groups III

Does the following condition imply amenability of $G$?

There exist constants $C<1$ and $\epsilon<1$ such that for every $S\subset G$ - finite set, there exists a finite $\widetilde{S}\subseteq S$ and $F \subset G$ - finite, such that

$|\widetilde{S}|\geq \epsilon |S|$

$|sF \Delta F|\leq C\cdot |F|$ for every $s\in \widetilde{S}$

Details on the question:

Note that, we have the following (see here):

G is amenable iff there exists $C<1$ such that for every $S\subset G$ - finite set, there exists $F \subset G$ - finite, such that

$|sF \Delta F|\leq C\cdot |F|$ for every $s\in S$

Also the existence of non-amenable group which has the property described here will give an example of the non-amenable groups which satisfies the condition of the present question. However, there is a strong evidence that there is no such non-amenable group, see here and here.

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