Uniform homogeneity

A mathematical structure is called homogeneous if every isomorphism between its ``small" substructures extends to an automorphism. If additionally, this extension can be made algebraic, namely, preserving compositions, we then say that the structure is uniformly homogeneous. Homogeneous structures are well known in model theory, where the meaning of ``small" is ``finite" or ``finitely generated". Fraisse theory provides basic tools for constructing or recognizing homogeneous structures. The aim of the talk is to present examples of homogeneous structures that are far from being uniformly homogeneous. One of our examples has the property that its automorphism group is torsion-free, while its finite substructures have non-trivial automorphisms. The results come from a joint work with S. Shelah.