2-dimensional Convexity Numbers and P4-free Graphs

Abstract

AbstractFor S ⊆ ℝn a set C ⊆ 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 ⊆ ℝ2 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 ℝ2 without uncountable 3-cliques in terms of clopen, P4-free graphs on Polish spaces.