A short note on the characterization of countable chains with finite big Ramsey spectra

Abstract: In this short note we confirm the deep structural correspondence between the complexity of a countable scattered chain (= strict linear order) and its big Ramsey combinatorics: we show that a countable scattered chain has finite big Ramsey degrees if and only if it is of finite Hausdorff rank. This also provides a complete characterization of countable chains whose big Ramsey spectra are finite. We expand the notion of big Ramsey spectrum to monomorphic structures and give a sufficient condition for a monomorphic countable structure to have finite big Ramsey spectrum.