Combinatorial complexity in o-minimal geometry
Saugata Basu · Proceedings of the London Mathematical Society · 2009
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. [Crossing patterns of semi-algebraic sets, J. Combin. Theory Ser. A 111 (2005), 310–326. MR 2156215 (2006k:14108)], originally proved for semi-algebraic sets of fixed description complexity to this more general setting.