Indiscernibles, EM-types, and Ramsey Classes of Trees

TL;DR

This paper extends the class of structures for which indiscernible sets have the modeling property, showing that certain classes of finite trees are Ramsey, by expanding to ordered, locally finite structures that isolate quantifier-free types.

Contribution

It broadens the known class of structures with the modeling property to include ordered, locally finite structures that isolate quantifier-free types, leading to new Ramsey results for finite trees.

Findings

Certain classes of finite trees are Ramsey

Expanded structures still have the modeling property

Generalization from ordered structures to locally finite structures

Abstract

It was shown in \cite{sc12} that for a certain class of structures $\I$, $\I$-indexed indiscernible sets have the modeling property just in case the age of $\I$ is a Ramsey class. We expand this known class of structures from ordered structures in a finite relational language to ordered, locally finite structures which isolate quantifier-free types by way of quantifier-free formulas. As a corollary, we may conclude that certain classes of finite trees are Ramsey, some previously known. See updated paper for new references.