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.