Automorphism Groups of Generic Structures: Extreme Amenability and Amenability

We investigate correspondences between extreme amenability and amenability of automorphism groups of Fra\"iss\'e-Hrushovski generic structures that are obtained from smooth classes, and their Ramsey type properties of their smooth classes, similar to Kechris, Pestov and Todorcevic, and Tatch Moore. In particular, we focus on some Fra\"iss\'e-Hrushovski generic structures that are obtained from pre-dimension functions. Using these correspondences, we prove that automorphism groups of ordered Hrushovski generic graphs are not extremely amenable in both cases of collapsed and uncollapsed. Moreover, we prove that automorphism groups of Fra\"iss\'e-Hrushovski generic structures that are obtained from pre-dimension functions with rational coefficients are not amenable.