Combinatorial complexity in O-minimal geometry

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 ominimal 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. [3], originally proved for semi-algebraic sets of fixed description complexity to this more general setting.

• Type: Conference
• Year: 2007
• Where: STOC
• Authors: Saugata Basu
• Keywords: Algorithms, Definable Sets, Semi-Pfaffian Sets, STOC 2007, Topological Complexity