  We study a class of generalized expansive dynamical systems for which at most
countable orbits can be accompanied by an arbitrary given orbit. Examples of
different levels of generalized expansiveness are constructed. When the
dynamical system is countable, a characterization of n-expansiveness is given
for any natural number n, and as a consequence examples of dynamical systems
with van der Waerden depth equal to any given countable ordinal are
demonstrated, which solves open questions existing in the literature.
