On Ramsey properties of classes with forbidden trees

TL;DR

This paper proves that classes of relational structures avoiding certain trees, when expanded with unary relations and a linear order, satisfy the Ramsey property, extending combinatorial principles to new classes.

Contribution

It introduces a method to expand classes of structures with forbidden trees to achieve the Ramsey property using unary relations and linear orderings.

Findings

Expanded classes with unary relations have the Ramsey property.

The approach applies to classes defined by forbidden relational trees.

The results extend Ramsey theory to new classes of relational structures.

Abstract

Let F be a set of relational trees and let Forbh(F) be the class of all structures that admit no homomorphism from any tree in F; all this happens over a fixed finite relational signature $\sigma$. There is a natural way to expand Forbh(F) by unary relations to an amalgamation class. This expanded class, enhanced with a linear ordering, has the Ramsey property.