Geodesic
From Ramsey degrees to Ramsey expansions via weak amalgamation
Theory and applications of categories, Tome 41 (2024), pp. 1513-1535.

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In this paper we provide another argument to support the recently reinvigorated interest in treating Ramsey-type phenomena categorically. Using purely categorical strategies we construct a Ramsey expansion for every category of finite objects with finite small Ramsey degrees. Our construction is based on the relationship between small Ramsey degrees, weak amalgamation, and recent results about weak Fraïssé categories. Starting from the fact that weak Fraïssé categories allow for certain model-theoretic properties to be reflected in the free ω-cocompletion of the category, we show that classes with finite Ramsey degrees have weak amalgamation and then invoke the machinery of weak Fraïssé categories to perform the construction. This improves previous similar results where an analogous construction was carried out under the assumption that everything sits comfortably in a bigger class with enough infrastructure, and that in this wider context there is an ultrahomogeneous structure under whose umbrella the construction takes place.
Publié le :
Classification : 18A35, 05C55
Keywords: Fraïssé categories, free ω-cocompletion, Ramsey degrees, Ramsey expansion, weak amalgamation 
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     author = {Dragan Ma\v{s}ulovi\'c and Andy Zucker},
     title = {From {Ramsey} degrees to {Ramsey} expansions via weak amalgamation},
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     pages = {1513--1535},
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     year = {2024},
     language = {en},
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Dragan Mašulović; Andy Zucker. From Ramsey degrees to Ramsey expansions via weak amalgamation. Theory and applications of categories, Tome 41 (2024), pp. 1513-1535. http://geodesic.mathdoc.fr/item/TAC_2024_41_a42/