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In this thesis we solve some open problems related to the homomorphism order of digraphs. We begin by introducing the basic concepts of graphs and homomorphisms and studying some properties of the homomorphism order of digraphs. Then we present the new results. First, we show that the class of digraphs containing cycles has the fractal property (strengthening the density property) . Then we show a density theorem for the class of proper oriented trees. Here we say that a tree is proper if it is not a path. Such result was claimed in 2005 but none proof have been published ever since. We also show that the class of proper oriented trees, in addition to be dense, has the fractal property. We end by considering the consequences of these results and the remaining open questions in this area., Outgoing

Given a countably infinite hypergraph $\mathcal R$ and a finite hypergraph $\mathcal A$, the \emph{big Ramsey degree} of $\mathcal A$ in $\mathcal R$ isthe 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)., Peer Reviewed, Postprint (published version)