From Ramsey degrees to Ramsey expansions via weak amalgamation (Q6634619)

It was observed in the late 1990s [\textit{W. L. Fouché}, Discrete Math. 167--168, 309--315 (1997; Zbl 0873.05076); J. Comb. Theory, Ser. A 85, No. 2, 135--147 (1999; Zbl 0918.05097); East-West J. Math. 1, No. 1, 43--60 (1998; Zbl 0914.05066)] that many concrete classes of finite structures where a Ramsey expansion had been identified also enjoys a weaker property of having finite Ramsey degrees. For Fraissé classes, it was established in [\textit{A. S. Kechris} et al., Geom. Funct. Anal. 15, No. 1, 106--189 (2005; Zbl 1084.54014)] that classes with a Ramsey expansion have finite Ramsey degrees, while the second author [Trans. Am. Math. Soc. 368, No. 9, 6715--6740 (2016; Zbl 1359.37024)] showed that small Ramsey degrees suffice for the existence of a Ramsey expansion. A more combinatorial proof of the same fact can be found in [\textit{L. N. Van Thé}, ``Finite Ramsey degrees and Fraïssé expansions with the Ramsey property'', Preprint, \url{arXiv:1705.10582}], and a reinterpretation of the latter proof in the language of category theory in [\textit{D. Mašulović}, Appl. Categ. Struct. 29, No. 1, 141--169 (2021; Zbl 1465.18002)]. All these proof make key use of the fact that the classes are Fraïssé, in the sense that every thing sits comfortably in a bigger class with enough infrastructure, and that in this wider context there is aan ultrahomogeneous structure under whose umbrella the construction takes place.\N\NThis paper aims to show that the assumption about the ambient class in which an ultrahomogeneous object oversees the construction is unnecessary. The authors construct a Ramsey expansion without imposing any additional assumptions for each category of finite objects with finite, small Ramsey degrees.\N\NThe synopsis of the paper goes as follows.\N\N\begin{itemize}\N\item[\S 2] quickly fixes some notation and conventions.\N\N\item[\S 3] shows (Theorem 3.2) that classes with finite Ramsey degrees have the weak amalgamation property, which is an analogue of \textit{J. Nešetřil} and \textit{V. Rödl}'s result [J. Comb. Theory, Ser. A 22, 289--312 (1977; Zbl 0361.05017)] claiming that the Ramsey property implies amalgamation. The authors then recalls from [\textit{W. Kubiś}, Ann. Pure Appl. Logic 165, No. 11, 1755--1811 (2014; Zbl 1329.18002); Theory Appl. Categ. 38, 27--63 (2022; Zbl 1502.18004)] how an ambient category and a weakly homogeneous object in it can be constructed from a category with weak amalgamation by taking the free \(\varpi\)-cocompletion of the original category.\N\N\item[\S 4] upgrades the results from [\textit{D. Mašulović}, Appl. Categ. Struct. 29, No. 1, 141--169 (2021; Zbl 1465.18002)] to show that if everything sits in a bigger category in which there is a weakly homogeneous and locally finite object universal for the expanded category, then there is a convenient expansion which can be trimmed down to a Ramsey expansion.\N\N\item[\S 5] establishes the main result of the paper.\N\NTheorem. Let \(\boldsymbol{C}\) be a directed category of finite objects whose morphisms are mono. Then \(\boldsymbol{C}\) has finite Ramsey degrees iff there exists a category \(\boldsymbol{C}^{\ast}\) with the Ramsey property and a reasonable precompact expansion with unique restrictions \(U:\boldsymbol{C}^{\ast }\rightarrow\boldsymbol{C}\) with the expansion property.\N\N\item[\S 6] As a corollary, the main result of the paper is specialized to arbitrary first-order structures with a dual result about the relationship of small dual Ramsey degrees and dual Ramsey expansions.\N\end{itemize}