  In this paper we prove tight bounds on the combinatorial and topological
complexity of sets defined in terms of $n$ definable sets belonging to some
fixed definable family of sets in an o-minimal structure. This generalizes the
combinatorial parts of similar bounds known in the case of semi-algebraic and
semi-Pfaffian sets, and as a result vastly increases the applicability of
results on combinatorial and topological complexity of arrangements studied in
discrete and computational geometry. As a sample application, we extend a
Ramsey-type theorem due to Alon et al., originally proved for semi-algebraic
sets of fixed description complexity to this more general setting.
