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 σ1,..., σ n and τ1 ofR (in the sense of Baire category), (R,σ1,...,σ n ), ≅ (R,τ1,...,τ 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.
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Work supported by a Sloan Fellowship and by NSF grant DMS-8903378.
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Hrushovski, E. Extending partial isomorphisms of graphs. Combinatorica 12, 411–416 (1992). https://doi.org/10.1007/BF01305233
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DOI: https://doi.org/10.1007/BF01305233