Fraisse classes with simply characterized big Ramsey degrees

TL;DR

This paper introduces a new property for Fraisse structures that ensures finite big Ramsey degrees with simple characterizations, unifying and simplifying Ramsey theory for certain classes of structures.

Contribution

It presents a strengthened property that guarantees finite big Ramsey degrees and indivisibility, with new coding tree formulations and direct degree characterizations.

Findings

Finite big Ramsey degrees characterized simply

Fraisse structures with the property are indivisible

Unified approach to Ramsey theory for specific classes

Abstract

We formulate a property strengthening the Disjoint Amalgamation Property and prove that every Fraisse structure in a finite relational language with relation symbols of arity at most two having this property has finite big Ramsey degrees which have a simple characterization. It follows that any such Fraisse structure admits a big Ramsey structure. Furthermore, we prove indivisibility for every Fraisse structure in an arbitrary finite relational language satisfying this property. This work offers a streamlined and unifying approach to Ramsey theory on some seemingly disparate classes of Fraisse structures. Novelties include a new formulation of coding trees in terms of 1-types over initial segments of the Fraisse structure, and a direct characterization of the degrees without appeal to the standard method of "envelopes".