Ordered asymptotic classes of finite structures

TL;DR

This paper introduces O-asymptotic classes of finite structures, combining asymptotic class concepts with o-minimality, and classifies their ultraproducts as superrosy of U-thorn-rank 1 and NTP2.

Contribution

It presents the first formal definition of O-asymptotic classes and provides a cell-decomposition theorem and a classification of their ultraproducts.

Findings

Ultraproducts are superrosy of U-thorn-rank 1

Ultraproducts are NTP2 (inp-minimal)

Cell-decomposition for O-asymptotic classes

Abstract

In this paper we introduce the concept of O-asymptotic classes of finite structures, melding ideas coming from 1-dimensional asymptotic classes and o-minimality. The results we present here include a cell-decomposition result for O-asymptotic classes and a classification of their infinite ultraproducts: Every infinite ultraproduct of structures in an O-asymptotic class is superrosy of U-thorn-rank 1, and NTP2 (in fact, inp-minimal).