On $n$-saturated closed graphs

TL;DR

This paper constructs, for every natural number n, an n-saturated closed graph on the Cantor space, completing the understanding of saturation properties of closed graphs in descriptive set theory.

Contribution

It proves the existence of n-saturated closed graphs on the Cantor space for all n, using inverse limits and probabilistic methods, filling a gap in the classification of such graphs.

Findings

Existence of n-saturated closed graphs for all n.

Construction via inverse limits of finite graphs.

Probabilistic argument used in the key lemma.

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.