On n -saturated Closed Graphs

Geschke proved in [ 2 ] that there is a clopen graph on 2^ω which is 3-saturated, but the clopen graphs on 2^ω do not even have infinite subgraphs that are 4-saturated; however there is F_σ graph that is ω _1 -saturated. It turns out that there is no closed graph on 2^ω which is ω -saturated, see [ 3 ]. In this note we complete this picture by proving that for every n∈ℕ there is an n -saturated closed graph on the Cantor space 2^ω . The key lemma is based on probabilistic argument. The final construction is an inverse limit of finite graphs.