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Combinatorial complexity in o-minimal geometry

Saugata Basu
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Abstract

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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Saugata Basu. 2009-05-01. Combinatorial complexity in o-minimal geometry. https://doi.org/10.1112/plms%2Fpdp031

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