Automorphism groups and Ramsey properties of sparse graphs

Jan Hubicka

We disucss automorphisms groups of the structures constructed by Hrushovski predimension construction. Following one of the open questions show that Hrushovski's example of an $\omega$-categorical sparse graph has no $\omega$-categorical expansion with extremely amenable automorphism group. We however give an explicit description of maximal amenable and maximal extremely amenable subgroups by combinatorial methods (showing an expansion with the extension property for partial automorphisms and another expansion which is a Ramsey class).