Asymptotic Classes of Trees and $\aleph_0$-categoricity

TL;DR

This paper characterizes $eth_0$-categorical theories of trees using tree plans and asymptotic classes, linking finite tree classes to ultraproducts and analyzing their model-theoretic properties.

Contribution

It introduces a novel characterization of $eth_0$-categorical theories of trees via tree plans and asymptotic classes, connecting finite and infinite models.

Findings

Asymptotic classes of finite trees produce $eth_0$-categorical ultraproducts.

Tree plans effectively characterize $eth_0$-categorical theories of trees.

Model-theoretic properties of these classes are systematically studied.

Abstract

This paper focuses on the characterization of $\aleph_0$-categorical theories of trees in the following sense: for any $\aleph_0$-cateorical theory $T$ of trees there is a tree plan $\Gamma$ such that $T=Th(\Gamma(\omega))$ where $\Gamma(\omega)$ is the generic model of a Fra\"iss\'e class $\mathbf{K}(\Gamma)$ obtained from the tree plan $\Gamma$. Also, it is shown that $\mathbf{K}(\Gamma)$ forms an asymptotic class, and its model-theoretic properties have been studied. Moreover, it is demonstrated that asymptotic classes of finite trees yield $\aleph_0$-categorical ultraproducts, and a characterization of supersimple, finite rank trees, using the notion of tree plan, is provided.