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Preprint Number 1851

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1851. Rebecca Coulson, Natasha Dobrinen, and Rehana Patel
The Substructure Disjoint Amalgamation Property implies big Ramsey structures

Submission date: 5 October 2020


We formulate a strengthening of the Disjoint Amalgamation Property and prove that every Fraisse class K in a finite relational language with this amalgamation property has finite big Ramsey degrees. Moreover, we characterize the exact degrees. It follows that the Fraisse structure of any class with this amalgamation property admits a big Ramsey structure. 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, essentially forcing on the structures themselves, and a direct characterization of the degrees without appeal to the standard method of “envelopes”, providing a clear analysis of the exact big Ramsey degrees.

Mathematics Subject Classification: 05D10, 05C55, 05C15, 05C05, 03C15, 03E75

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Full text arXiv 2010.02034: pdf, ps.

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