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2-dimensional Convexity Numbers and P4-free Graphs

Published online by Cambridge University Press:  20 November 2018

Stefan Geschke
Stefan Geschke*
Affiliation:
Hausdorff Center for Mathematics, Endenicher Allee 62, 53115 Bonn, Germany e-mail: stefan.geschke@hcm.uni-bonn.de
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Abstract

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For $S\,\subseteq \,{{\mathbb{R}}^{n}}$ a set $C\,\subseteq \,S$ is an $m$-clique if the convex hull of no $m$-element subset of $C$ is contained in $S$. We show that there is essentially just one way to construct a closed set $S\,\subseteq \,{{\mathbb{R}}^{2}}$ without an uncountable 3-clique that is not the union of countably many convex sets. In particular, all such sets have the same convexity number; that is, they require the same number of convex subsets to cover them. The main result follows from an analysis of the convex structure of closed sets in ${{\mathbb{R}}^{2}}$ without uncountable 3-cliques in terms of clopen, ${{P}_{4}}$-free graphs on Polish spaces.

Keywords

52A1003E1703E75convex coverconvexity numbercontinuous coloringperfect graphcograph
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 57 , Issue 1 , 14 March 2014 , pp. 61 - 71
Copyright
Copyright © Canadian Mathematical Society 2014

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