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Proceedings of the London Mathematical Society Advance Access published online on August 9, 2009
Proceedings of the London Mathematical Society, doi:10.1112/plms/pdp031
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© 2009 London Mathematical Society

Combinatorial complexity in o-minimal geometry

Saugata Basu

Department of Mathematics
Purdue University
West Lafayette, IN 47906
USA

Received 16 October 2007. Revision received 27 May 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.

2000 Mathematics Subject Classification 52C45.

The author was supported in part by an NSF grant CCF-0634907.


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