Dual Ramsey degrees for some classes of relational structures

Aleksa Džuklevski, Dragan Mašulović

Published 2024-07-04Version 1

In this paper we show that some natural classes of structures such as graphs, posets and metric spaces have both dual small and dual big Ramsey degrees with respect to some natural classes of morphisms such as quotient maps in case of graphs and posets, or non-expansive surjections in case of metric spaces. The only exception is the class of reflexive tournaments: they have neither dual small nor dual big Ramsey degrees. Our proof strategy is based on the categorical interpretation of structural Ramsey theory. Starting from a category we are interested in, we engineer an synthetic expansion to piggyback on a category where the dual Ramsey property has been established. We then use the additive properties of dual Ramsey degrees to get back to the original category of "natural" objects and morphisms and conclude that it has finite dual Ramsey degrees.