On Ramsey degrees, compactness and approximability (2406.02007v4)

Abstract: One of the consequences of the Compactness Principle in structural Ramsey theory is that the small Ramsey degrees cannot exceed the corresponding big Ramsey degrees, thereby justifying the choice of adjectives. However, it is unclear what happens in the realm of dual Ramsey degrees due to the lack of the compactness argument that applies to that setting. In this paper we present a framework within which both "direct" and dual Ramsey statements can be stated and reasoned about in a uniform fashion. We introduce the notion of approximability which yields a general compactness argument powerful enough to prove statements about both "direct" and dual Ramsey phenomena. We conclude the paper with an application of the new strategies by generalizing Voigt's $\star$-version of the Infinite Ramsey Theorem to a large class of relational structures and deriving a Ramsey statement for "loose colorings" of enumerated Fra\"{\i}ss\'{e} limits.