What I've learned about homogeneity

The classical notion of homogeneity in relational structures is about the possibility of extending any local isomorphism (i.e., an isomorphism with finite domain) to an automorphism of the potentially infinite structure. The original Fraïssé theorem establishes a correspondence between homogeneous countable structures and hereditary classes of finite structures with the joint embedding property and the amalgamation property. In this talk, I will present variations on the idea of homogeneity related to the extension of local homomorphisms of different types which can be extended to elements of submonoids of the endomorphism monoid of an infinite structure and explain their analogues of Fraïssé's theorem. If there is enough time, I will also present some structural results about countably infinite homomorphism-homogeneous graphs.