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Description

Each of the eight kinds of Ramsey degrees ($\{$small, big$\} \times \{$embedding, structural$\} \times \{$``direct'', dual$\}$) involves coloring a different kind of entity: substructures (for structural Ramsey degrees), embeddings (for embedding Ramsey degrees), quotients (for dual small structural Ramsey degrees), quotient maps (for dual small “embedding” Ramsey degrees), Borel sets of quotients (for dual big structural Ramsey degrees), or Borel sets of quotient maps (for dual big “embedding” Ramsey degrees). Still, the intuition suggests, and the overall beauty of mathematics demands, that there should really be just two: small and big. The dual versions ought to come for free from the Duality Principle of Category Theory. In this talk, I will present a framework where this simplification indeed takes place: categories of structures enriched over the category of measurable spaces and measurable maps. Although the description is a mouthful, the idea is quite simple: each hom-set carries a $\sigma$-algebra turning it into a measurable space, and the composition of morphisms is measurable. Then small and big Ramsey degrees are defined in the usual way, but taking into account only measurable colorings. I will focus on a convenient class of ``semi-standard'' measurable spaces. Intuitively, these can be seen as ``standard measurable spaces with fat atoms.'' When applied to categories of finite structures, where hom-sets are finite and the corresponding $\sigma$-algebras reduce to finite algebras, we recover all the familiar kinds of small Ramsey degrees, and uncover a wealth of new ones. When we extend the picture to include countably infinite structures, all forms of big Ramsey degrees reappear, along with many new examples. To prove the relationship between small and big ``direct'' Ramsey degrees we need the Compactness Principle, which breaks down in the dual setting. To restore the balance, I introduce a generalization and show that every small Ramsey degree ($\{$embedding, structural$\} \times \{$``direct'', dual$\}$, or some of the new ones) is indeed smaller than the corresponding big one, so the Universe remains in order. While this approach settles several conceptual (and admittedly aesthetic) questions, it also opens up new challenges, which is where I will end the talk.