4 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 using parameter spaces
We show that the universal homogeneous partial order has finite big Ramsey
degrees and discuss several corollaries. Our proof uses parameter spaces and
the Carlson-Simpson theorem rather than (a strengthening of) the
Halpern-L\"auchli theorem and the Milliken tree theorem, which are the primary
tools used to give bounds on big Ramsey degrees elsewhere (originating from
work of Laver and Milliken).
This new technique has many additional applications. To demonstrate this, we
show that the homogeneous universal triangle-free graph has finite big Ramsey
degrees, thus giving a short proof of a recent result of Dobrinen.Comment: 19 pages, 2 figure