Convex Geometry, Continuous Ramsey Theory, and Ideals Generated by Graphs of Functions

This is a survey of the results in [1, 2, 3, 4, 5, 6] on convexity numbers of closed sets in Rn, homogeneity numbers of continuous colorings on Polish spaces and families of functions covering Rn. 1. Convexity numbers of closed sets in R A natural way to measure the degree of non-convexity of a subset S of a real vector space is the convexity number γ(S), the least size of a family C of convex sets such that ⋃ C = S. We will almost exclusively consider closed subsets of R. Moreover, we concentrate on sets whose convexity numbers are uncountable. Such sets will be called uncountably convex. Sets with a countable convexity number are countably convex. A simple reason for a set S ⊆ R to be uncountably convex is the existence of an uncountable m-clique C ⊆ S for some m ∈ ω. C ⊆ S is an m-clique if for all F ∈ [C] the convex hull of F is not a subset of S. It can be shown that every closed set S ⊆ R that has an uncountable m-clique for some m in fact has a (nonempty) perfect m-clique. In particular, we have γ(S) = 2א0 for such a set. Let us point out that Caratheodory’s theorem implies that a subset of R has an uncountable m-clique for some m ∈ ω iff it has an uncountable (n+ 1)-clique. Therefore, if we are speaking about subsets of R, clique will always mean (n+ 1)-clique. It is natural to ask whether the existence of a perfect clique is the only reason for a closed set S ⊆ R to have γ(S) = 2א0 . Indeed, closed subsets of R are either countably convex or have a perfect clique. But this already fails in dimension 2. Kubís constructed a closed subset of R that does not have an uncountable clique but is still uncountably convex [6, Section 2, Theorem 1]. In particular, under CH the convexity number of the Kubís set is 2א0 . From the construction it is clear that the convexity number of the Kubís set can be characterized as follows. For {x, y} ∈ [ω] let ∆(x, y) = min{n ∈ ω : x(n) 6= y(n)}.