Combinatorial complexity in o-minimal geometry
Combinatorics
2014-02-26 v2 Logic
Abstract
In this paper we prove tight bounds on the combinatorial and topological complexity of sets defined in terms of 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.
Cite
@article{arxiv.math/0612050,
title = {Combinatorial complexity in o-minimal geometry},
author = {Saugata Basu},
journal= {arXiv preprint arXiv:math/0612050},
year = {2014}
}
Comments
25 pages. Revised version. To appear in the Proc. London Math. Soc