  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.
