Automorphism groups of generic structures: extreme amenability and amenability
This paper presents some correspondences between (extreme) amenability of automorphism groups of Fraïssé–Hrushovski generic structures, and Ramsey type properties of their smooth classes, similar to results of Kechris, Pestov and Todorcevic (2005) and Moore (2013). In particular, we focus on some Fraïssé–Hrushovski generic structures that are obtained from the pre-dimension functions $\delta _\alpha $ for $\alpha \geq 1$. Using these correspondences, it is shown that the automorphism groups of ordered Hrushovski generic graphs are not extremely amenable in both cases of collapsed and uncollapsed structures. Moreover, when $\alpha $ is rational we prove that the automorphism groups of Fraïssé–Hrushovski generic structures that are obtained from the pre-dimension functions $\delta _\alpha $ are not amenable.