26 research outputs found
Big Ramsey degrees of 3-uniform hypergraphs
Given a countably infinite hypergraph and a finite hypergraph
, the big Ramsey degree of in is the
least number such that, for every finite and every -colouring of the
embeddings of to , there exists an embedding from
to such that all the embeddings of to
the image have at most different colours.
We describe the big Ramsey degrees of the random countably infinite 3-uniform
hypergraph, thereby solving a question of Sauer. We also give a new
presentation of the results of Devlin and Sauer on, respectively, big Ramsey
degrees of the order of the rationals and the countably infinite random graph.
Our techniques generalise (in a natural way) to relational structures and give
new examples of Ramsey structures (a concept recently introduced by Zucker with
applications to topological dynamics).Comment: 8 pages, 3 figures, extended abstract for Eurocomb 201
Big Ramsey degrees and infinite languages
This paper investigates big Ramsey degrees of unrestricted relational
structures in (possibly) infinite languages. While significant progress has
been made in studying big Ramsey degrees, many classes of structures with
finite small Ramsey degrees still lack an understanding of their big Ramsey
degrees. We show that if there are only finitely many relations of every arity
greater than one, then unrestricted relational structures have finite big
Ramsey degrees, and give some evidence that this is tight. This is the first
time that finiteness of big Ramsey degrees has been established for an
infinite-language random structure. Our results represent an important step
towards a better understanding of big Ramsey degrees for structures with
relations of arity greater than two.Comment: 21 pages. An updated version strengthening the statement of the
positive results and fixing a mistake in the earlier version of the negative
result which now needs an extra assumptio