In this paper, \(L\) is a finite relational language, and the author considers countable (possible finite) \(L\)-structures which are homogeneous: that is, any isomorphism between finite substructures extends to an automorphism. The author surveys the structure theory (developed by the author and collaborators during the early 1980’s) for finite homogeneous \(L\)-structures and for stable countable homogeneous \(L\)-structures (the latter are limits of sequences of the former). Much of the theory was developed in the author’s paper “On countable stable structures which are homogeneous for a finite relational language” [Isr. J. Math. 49, 69-153 (1984; Zbl 0605.03013)]. Essentially, once \(L\) is fixed, the stable (possibly finite) countable homogeneous \(L\)-structures, if larger than a certain computable size, fall into finitely many families, such that in each family each structure is determined up to isomorphism by a bounded number of dimensions, which vary independently.
For the entire collection see [Zbl 0868.00035].