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2-dimensional Convexity Numbers and $P_4$-free Graphs

  Published:2012-09-21
 Printed: Mar 2014
  • Stefan Geschke,
    Hausdorff Center for Mathematics, Endenicher Allee 62, 53115 Bonn, Germany
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Abstract

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: convex cover, convexity number, continuous coloring, perfect graph, cograph convex cover, convexity number, continuous coloring, perfect graph, cograph
MSC Classifications: 52A10, 03E17, 03E75 show english descriptions Convex sets in $2$ dimensions (including convex curves) [See also 53A04]
Cardinal characteristics of the continuum
Applications of set theory
52A10 - Convex sets in $2$ dimensions (including convex curves) [See also 53A04]
03E17 - Cardinal characteristics of the continuum
03E75 - Applications of set theory
 

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