2023/05/04 by Barbosa, Keegan Dasilva, Mašulović, Dragan, Nenadov, Rajko
#Combinatorics (math.CO) #FOS: Mathematics #Logic (math.LO)
paper · doi:10.48550/arxiv.2305.02648
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.