Fraisse Structures with SDAP+, Part I: Indivisibility

TL;DR

This paper proves that Fraisse structures with the SDAP+ property are indivisible, introducing new coding tree formulations and using forcing methods for the proof, advancing Ramsey theory in relational structures.

Contribution

It provides a new formulation of coding trees and a direct forcing-based proof of indivisibility for Fraisse structures with SDAP+.

Findings

Fraisse structures with SDAP+ are indivisible.

New coding tree formulation in terms of 1-types.

Forcing method used for unbounded searches in proofs.

Abstract

This is Part I of a two-part series regarding Ramsey properties of Fraisse structures satisfying a property called SDAP+, which strengthens the Disjoint Amalgamation Property. We prove that every Fraisse structure in a finite relational language with relation symbols of any finite arity satisfying this property is indivisible. Novelties include a new formulation of coding trees in terms of 1-types over initial segments of the Fraisse structure, and a direct proof of indivisibility which uses the method of forcing to conduct unbounded searches for finite sets. 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 a theorem from Part I as a pigeonhole principle for induction arguments. This work offers a streamlined and unifying approach to Ramsey theory on some seemingly disparate classes of Fraisse structures.