A nonempty subset $S$ of a group $G$ is called *small* if there is an infinite sequence of elements $g_n$ in $G$ such that the translated sets $g_nS$ are pairwise disjoint.

Question: Is there a group which is a (disjoint) union of three small subsets, but it is not a union of two small subsets?

Remark: Such a group must be non-amenable (clear) and must not contain a copy of the non-abelian free group (in fact it is an exercise to see that the groups which are a union of two small sets are exactly those containing the free group).

Bonus question: Is every non-amenable group a finite union of small subsets?

proofof paradoxality for nonamenable groups rather than merely from its statement? (If you did not, it might be worth trying.) Also, maybe you should directly ask Grigorchuk, he might know some interesting examples. $\endgroup$1more comment