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

Combinatorics;Discrete Mathematics;Logic