Finite big Ramsey degrees in universal structures

Big Ramsey degrees of finite structures are usually considered with respect to a Frai sse limit. Building mainly on the work of Devlin, Sauer, Laflamme and Van The, in this paper we consider structures which are not Frai sse limits, and still have the property that their finite substructures have finite big Ramsey degrees in them. For example, the class of all finite acyclic oriented graphs is not a Frai sse class, and yet we show that there is a countably infinite acyclic oriented graph in which every finite acyclic oriented graph has finite big Ramsey degree. Our main tools come from category theory as it has recently become evident that the Ramsey property is not only a deep combinatorial property, but also a genuine categorical property.

[1]  Dragan Masulovic.  A Dual Ramsey Theorem for Permutations , 2017, Electron. J. Comb..

[2]  Dragan Masulovic,et al.  Countable homogeneous linearly ordered posets , 2012, Eur. J. Comb..

[3]  Frank Plumpton Ramsey,et al.  On a Problem of Formal Logic , 1930 .

[4]  R. Fraïssé.  Sur l'extension aux relations de quelques propriétés des ordres , 1954 .

[5]  Norbert Sauer.  COLORING SUBGRAPHS OF THE RADO GRAPH , 2006, Comb..

[6]  Wacław Sierpiński,et al.  Sur un espace métrique séparable universel , 1945 .

[7]  Robert Gray,et al.  Countable connected-homogeneous graphs , 2010, J. Comb. Theory, Ser. B.

[8]  L. Nguyen.  Big Ramsey degrees and divisibility in classes of ultrametric spaces , 2007 .

[9]  Claude Laflamme,et al.  Partition properties of the dense local order and a colored version of Milliken’s theorem , 2010, Comb..

[10]  Natasha Dobrinen.  The Ramsey theory of the universal homogeneous triangle-free graph , 2020, J. Math. Log..

[11]  Wilfrid Hodges,et al.  Model Theory: The existential case , 1993 .

[12]  P. Erdos,et al.  Asymmetric graphs , 1963 .

[13]  Dragan Masulovic.  Pre-adjunctions and the Ramsey property , 2018, Eur. J. Comb..

[14]  Peter J. Cameron,et al.  Homogeneous Permutations , 2002, Electron. J. Comb..

[15]  N.W. Sauer.  Distance Sets of Urysohn Metric Spaces , 2013, Canadian Journal of Mathematics.

[16]  V. Pestov,et al.  Fraïssé Limits, Ramsey Theory, and topological dynamics of automorphism groups , 2003 .

[17]  Maurice Pouzet,et al.  Divisibility of countable metric spaces , 2007, Eur. J. Comb..

[18]  Jaroslav Nešetřil.  Ramsey theory , 1996 .

[19]  Vaught,et al.  A family of countable homogeneous graphs , 1971 .

[20]  Willem L. Fouché,et al.  Symmetry and the Ramsey Degrees of Finite Relational Structures , 1999, J. Comb. Theory, Ser. A.

[21]  Jirí Adámek,et al.  Abstract and Concrete Categories - The Joy of Cats , 1990 .

[22]  W. Fouché.  Symmetry and the Ramsey degree of posets , 1997, Discret. Math..

[23]  James H. Schmerl,et al.  Countable homogeneous partially ordered sets , 1979 .

[24]  Andy Zucker.  Big Ramsey degrees and topological dynamics , 2018, Groups, Geometry, and Dynamics.

[25]  Hans Jürgen Prömel.  Induced Partition Properties of Combinatorial Cubes , 1985, J. Comb. Theory, Ser. A.

[26]  Lynn Scow,et al.  Categorical equivalence and the Ramsey property for finite powers of a primal algebra , 2017 .