1. CMB 2012 (vol 57 pp. 61)
|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, cograph
Categories:52A10, 03E17, 03E75
2. CMB 2010 (vol 53 pp. 286)
|Orders of π-Bases|
We extend the scope of B. Shapirovskii's results on the order of $\pi$-bases in compact spaces and answer some questions of V. Tkachuk.
Keywords:Shapirovskii π-base, point-countable π-base, free sequences, canonical form for ordinals
Categories:54A25, 03E10, 03E75, 54A35