Convex Ramsey matrices and non-amenability of automophism groups of generic structures

In this paper we prove that the automorphism groups of certain countable generic structures are not amenable. For doing that, we first prove the existence of particular matrices that do not satisfy the convex Ramsey condition. For a pair of elements in a smooth class, we introduce the property of forming a free-pseudoplane in the generic structure. We then prove the non-amenability of the automorphism group of a generic structure obtained from a smooth class with a pair that forms a free-pseudoplane. As an application we show that the automorphism group of an ab-initio generic structure that is constructed using a pre-dimension function with irrational coefficients is not amenable.