Combinatorial complexity in o-minimal geometry

Details Combinatorial complexity in o-minimal geometry Combinatorial complexity in o-minimal geometry

2006-12-02

arxiv.org/abs/math/0612050v2

Proc. London Math. Soc, (3) 100 (2010) 405-428

Mathematics

Combinatorics

25 pages. Revised version. To appear in the Proc. London Math. Soc

Scientific paper

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.

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