Let \(\mathcal{M}\) be a Fraïssé structure (eg the random graph), and let \(\mathcal{A}\) be a countably infinite structure which is embeddable in \(\mathcal{M}\). If \(\mathcal{M}\) has free amalgamation, then there exists a Katetov embedding of \(\mathcal{A}\) into \(\mathcal{M}\): an embedding such that each automorphism of \(\mathcal{A}\) extends to an automorphism of \(\mathcal{M}\). Is this embedding "common" or "uncommon"?

To answer this, we investigate generic embeddings of \(\mathcal{A}\) into \(\mathcal{M}\). An embedding of \(\mathcal{A}\) into \(\mathcal{M}\) is said to be generic if it lies in a comeagre set inside the Polish space Emb(\(\mathcal{A}\), \(\mathcal{M}\)).

We will answer the following three questions:

- When are two embeddings of \(\mathcal{A}\) into \(\mathcal{M}\) generically isomorphic via an automorphism of \(\mathcal{M}\)?

- When is \(\mathcal{A}\) generically corigid (i.e. \(\mathrm{Aut}(\mathcal{M}/\mathcal{A}\)) trival)?

- Let \(\mathcal{\gamma}\) lie in \(\mathrm{Aut}(\mathcal{A}\)). When is \(\mathcal{\gamma}\) generically isomorphic via an automorphism of \(\mathcal{M}\)?

We will also discuss a wide range of examples in the context of these three questions.