Simple structures axiomatized by almost sure theories

TL;DR

This paper classifies simple, omega-categorical structures with SU-rank 1 and trivial pregeometry, linking their properties to almost sure theories of finite structures and extension properties.

Contribution

It provides a classification of certain simple structures via extension properties and their approximation by almost sure theories, highlighting conditions for stability and minimality.

Findings

Structures satisfy specific extension properties.

They can be approximated by almost sure theories of finite structures.

Attributes almost surely true lead to omega-stability or strong minimality.

Abstract

In this article we give a classification of the binary, simple, $\omega$-categorical structures with SU-rank 1 and trivial pregeometry. This is done both by showing that they satisfy certain extension properties, but also by noting that they may be approximated by the almost sure theory of some sets of finite structures equipped with a probability measure. This study give results about general almost sure theories, but also considers certain attributes which, if they are almost surely true, generate almost sure theories with very specific properties such as $\omega$-stability or strong minimality.