Infinite-dimensional Ramsey theory for binary free amalgamation classes

Abstract

We develop infinite-dimensional Ramsey theory for Fraïssé limits of finitely constrained free amalgamation classes in finite binary languages. We show that our approach is optimal and in particular, recovers the exact big Ramsey degrees proved in [2] for these structures. A crucial step in the work develops the new notion of an A.3(2)-ideal and shows that Todorcevic's Abstract Ramsey Theorem [34] holds when Axiom A.3(2) is replaced by the weaker assumption of an A.3(2)-ideal.

Big Ramsey Degrees of the Generic Partial Order

“…The proof of the Ramsey theorem for coding trees develops set-theoretic forcing on such trees, building on Harrington's forcing proof of the Halpern-Läuchli theorem. Coding trees have been generalized to free amalgamation classes in binary languages in a series of papers [BCD + 23b,Dob23,DZ23,Zuc22]. Recently, Hubička [Hub20] introduced a new connection between the Carlson-Simpson theorem [CS84] and big Ramsey degrees, which has significantly simplified some of earlier results and made it possible to bound big Ramsey degrees of partial orders and metric spaces [BCH + 21a, BCD + 23a].…”