Given a countably infinite hypergraph R and a finite hypergraph
A, the big Ramsey degree of A in R is the
least number L such that, for every finite k and every k-colouring of the
embeddings of A to R, there exists an embedding f from
R to R such that all the embeddings of A to
the image f(R) have at most L 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