Canadian Mathematical Society www.cms.math.ca
 location:  Publications → journals
Search results

Search: All articles in the CMB digital archive with keyword convexity number

 Expand all        Collapse all Results 1 - 1 of 1

1. CMB 2012 (vol 57 pp. 61)

Geschke, Stefan
 2-dimensional Convexity Numbers and \$P_4\$-free Graphs 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, cographCategories:52A10, 03E17, 03E75

© Canadian Mathematical Society, 2014 : https://cms.math.ca/