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On $n$-saturated closed graphs

Geschke proved that there is 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. In this note we complete this picture by proving that for every $n$ there is an $n$-saturated closed graph on the Cantor space $2^ω$. The key lemma is based on probabilistic argument. The final construction is an inverse limit of finite graphs.