  We explore recurrence properties arising from dynamical approach to
combinatorial problems like the van der Waerden Theorem. We describe relations
between these properties, study their consequences for dynamics, and
demonstrate connections to combinatorial problems. In particular, we present a
measure-theoretical analog of a result of Glasner on multi-transitivity in
topological weakly mixing minimal maps. We also obtain a dynamical proof of the
existence of a C-set with zero Banach density.
