## Clopen graphs

### Volume 220 / 2013

Fundamenta Mathematicae 220 (2013), 155-189
MSC: Primary 05C63; Secondary 03E15, 03E17.
DOI: 10.4064/fm220-2-5

#### Abstract

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 $X^2$ 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 quasi-orders of clopen graphs on compact zero-dimensional metric spaces with topological or combinatorial embeddability are Tukey equivalent to $\omega^\omega$ with eventual domination. In particular, the dominating number $\mathfrak d$ is the least size of a family of clopen graphs on compact metric spaces such that every clopen graph on a compact zero-dimensional metric space embeds into a member of the family. We also show that there are $\aleph_0$-saturated clopen graphs on $\omega^\omega$, while no $\aleph_1$-saturated graph embeds into a clopen graph on a Polish space. There is, however, an $\aleph_1$-saturated $F_\sigma$ graph on $2^\omega$.