A survey of homogeneous structures

A relational first order structure is homogeneous if it is countable (possibly finite) and every isomorphism between finite substructures extends to an automorphism. This article is a survey of several aspects of homogeneity, with emphasis on countably infinite homogeneous structures. These arise as Fraissé limits of amalgamation classes of finite structures. The subject has connections to model theory, to permutation group theory, to combinatorics (for example through combinatorial enumeration, and through Ramsey theory), and to descriptive set theory. Recently there has been a focus on connections to topological dynamics, and to constraint satisfaction. The article discusses connections between these topics, with an emphasis on examples, and on special properties of an amalgamation class which yield important consequences for the automorphism group.