Abstract
Given a countably infinite hypergraph R and affnite 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).