Fraisse structures with SDAP+, Part II: Simply characterized big Ramsey 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.
Cite
@article{arxiv.2207.06505,
title = {Fraisse structures with SDAP+, Part II: Simply characterized big Ramsey structures},
author = {Rebecca Coulson and Natasha Dobrinen and Rehana Patel},
journal= {arXiv preprint arXiv:2207.06505},
year = {2022}
}
Comments
At the request of the referee, our preprint arXiv:2010.02034 has been divided into two papers, Fraisse structures with SDAP+, Parts I and II. The property SDAP+ in the original preprint has been divided into a simpler version, again called SDAP+, and a labeled version called LSDAP+. All results in the preprint 2010.02034 remain the same in Parts I and II. arXiv admin note: text overlap with arXiv:2207.06393