  Liouville property of actions of discrete groups can be reformulated in terms
of existence co-F$\o$lner sets. Since every action of amenable group is
Liouville, the property can be served as an approach for proving
non-amenability. The verification of this property is conceptually different
than finding a non-amenable action. There are many groups that are defined by
strongly transitive actions. In some cases amenability of such groups is an
open problem. We define $n$-Liouville property of action to be Liouville
property of point-wise action of the group on the sets of cardinality $n$. We
reformulate $n$-Liouville property in terms of additive combinatorics and prove
it for $n=1, 2$. The case $n\geq 3$ remains open.
