Fraisse structures with SDAP+, Part II: Simply characterized big Ramsey structures

TL;DR

This paper establishes that certain Fraisse structures with SDAP+ property and relation symbols of arity at most two have finite big Ramsey degrees with a simple characterization, enabling a unified approach to their Ramsey properties.

Contribution

It proves finite big Ramsey degrees for a class of Fraisse structures with SDAP+ and provides a direct, envelope-free characterization of these degrees.

Findings

Fraisse structures with SDAP+ have finite big Ramsey degrees.

A simple characterization of these degrees is achieved.

The approach unifies Ramsey theory for various Fraisse structures.

Abstract

This is Part II of a two-part series regarding Ramsey properties of Fraisse structures satisfying a property called SDAP+, which strengthens the Disjoint Amalgamation Property. In Part I, we prove that every Fraisse structure in a finite relational language with relation symbols of any finite arity satisfying this property is indivisible. In Part II, we 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. Part II utilizes the notion of coding trees of 1-types developed in Part I and a theorem from Part I which functions as a pigeonhole principle for induction arguments in this paper. Our approach yields a direct characterization of the degrees without appeal to the standard method of "envelopes". This work offers a streamlined and unifying approach to Ramsey theory on some seemingly disparate classes of Fraisse structures.