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

Amenability, proximality and higher order syndeticity