Clopen Graphs, Inverse Limits, and Cochromatic Numbers

A graph G on a topological space X as its set of vertices is clopen if the edge relation of G is a clopen subset of X2 without the diagonal. We study clopen graphs on Polish spaces in terms of their finite induced subgraphs and obtain information about their cochromatic numbers. In this context we investigate modular profinite graphs, a class of graphs obtained from finite graphs by taking inverse limits. This continues the investigation of continuous colorings on Polish spaces and their homogeneity numbers started in [11] and [9]. We show that clopen graphs on compact spaces have no infinite induced subgraphs that are 4-saturated. In particular, there are countably infinite graphs such as Rado’s random graph that do not embed into any clopen graph on a compact space. Using similar methods, we show that the dominating number d is the least size of a family of clopen graphs on compact metric spaces such that every clopen graph on a compact metric space embeds into a member of the family. 1. Continuous colorings and clopen graphs Definition 1. An n-coloring with k colors on a set X is a function c : [X] → k. We are only interested in 2-colorings and among those we are mostly interested in colorings with 2 colors. Hence we call 2-colorings with 2 colors just colorings. If X is a topological space and k ∈ ω, then a coloring c : [X] → k is continuous if for all {x, y} ∈ [X] there are disjoint open sets U, V ⊆ X with x ∈ U and y ∈ V such that for all a ∈ U and all b ∈ V , c(x, y) = c(a, b). This is just continuity with respect to the natural topology on [X]. If c is a coloring on X and d is a coloring on Y , we write c ≤ d if there is a topological embedding e : X → Y , i.e., a homeomorphism onto its image, that preserves colors in the sense that for all distinct x0, x1 ∈ X we have c(x0, x1) = d(e(x0, x1)). A graph G is a set V (G) of vertices together with a set E(G) ⊆ [V (G)] of edges. G is a graph on X if V (G) = X. A graph G on a topological space is open, closed, Date: December 30, 2011.