Extending partial isomorphisms of graphs

Theorem Let X be a finite graph. Then there exists a finite graph Z containing X as an induced subgraphs, such that every isomorphism between induced subgraphs of X extends to an automorphism of Z.The graphZ may be required to be edge-transitive. The result implies that for anyn, there exists a notion of a genericn-tuple of automorphism of the Rado graphR (the random countable graph): for almost all automorphism ₁,..., _n and ₁ ofR (in the sense of Baire category), (R,₁,...,_n), (R,₁,...,_n). The problem arose in a recent paper of Hodges, Hodgkinson, Lascar and Shelah, where the theorem is used to prove the small index property forR.Work supported by a Sloan Fellowship and by NSF grant DMS-8903378.