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Many (omega)-categorical Structures Have the Small Index Property

Bernhard Herwig
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Abstract

A class K of finite structures is said to have the extension property for automorphisms (EP) if for every A in K there exists an extension B in K such that every partial isomorphism on the structure A extends to an automorphism of B. Hrushovski proved EP for the class of all finite graphs. The main problem is to keep B finite. Hodges, Hodkinson, Lascar and Shelah showed in their paper that in certain cases the EP for the class K implies the Small Index Property (SIP) for the "generic" countable structure determined by K. E.g. Hrushovskis result yields the SIP for the random graph. In this preprint we prove the EP for the class of all finite K_n - free graphs (i.e. graphs with no complete subgraph of given size n), which implies SIP for the generic K_n - free graph. Also we prove EP for a certain family of classes of directed graphs, which implies SIP for the "Henson digraphs". This is a family of continuum many non isomorphic countable directed omega-categorical graphs. Finally we state EP for a more general family of classes, which covers all the cases mentioned above.

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Bernhard Herwig. 1995-02-14. Many (omega)-categorical Structures Have the Small Index Property. https://arxiv.org/abs/math/9502206

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