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2-dimensional Convexity Numbers and P4-free Graphs
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- Journal:
- Canadian Mathematical Bulletin / Volume 57 / Issue 1 / 14 March 2014
- Published online by Cambridge University Press:
- 20 November 2018, pp. 61-71
- Print publication:
- 14 March 2014
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- Article
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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.
