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

TL;DR

This paper demonstrates that automorphism groups of certain generic structures are non-amenable by constructing specific matrices and introducing the free-pseudoplane property, with implications for structures built via irrational pre-dimension functions.

Contribution

It introduces the concept of free-pseudoplane in smooth classes and proves non-amenability of automorphism groups for structures with this property.

Findings

Existence of matrices violating convex Ramsey condition.

Non-amenability of automorphism groups of structures with free-pseudoplane.

Non-amenability of automorphism groups of structures with irrational pre-dimension functions.

Abstract

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.