On $n$-saturated closed graphs

Abstract

Geschke proved that there is clopen graph on $2^{\omega }$ which is 3-saturated, but the clopen graphs on $2^{\omega }$ do not even have infinite subgraphs that are 4-saturated; however there is $F_{\sigma }$ graph that is ${\omega }_1$-saturated. It turns out that there is no closed graph on $2^{\omega }$ which is $\omega$-saturated. In this note we complete this picture by proving that for every $n$ there is an $n$-saturated closed graph on the Cantor space $2^{\omega }$. The key lemma is based on probabilistic argument. The final construction is an inverse limit of finite graphs.

5 pages