2022/07/13 by Rebecca Coulson, Coulson, Rebecca, Natasha Dobrinen +3
Computer Science · Mathematics · #Advanced Topology and Set Theory #Combinatorics (math.CO) #Computability, Logic, AI Algorithms #FOS: Mathematics #Limits and Structures in Graph Theory
paper · pdf · doi:10.48550/arxiv.2207.06393
openalex publication_date 2022/07/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.