  We show that the universal minimimal proximal flow and the universal minimal
strongly proximal flow of a discrete group can be realized as the Stone spaces
of translation invariant Boolean algebras of subsets of the group satisfying a
higher order notion of syndeticity. We establish algebraic, combinatorial and
topological dynamical characterizations of these subsets that we use to obtain
new necessary and sufficient conditions for strong amenability and amenability.
We also characterize dense orbit sets, answering a question of Glasner,
Tsankov, Weiss and Zucker.
