@MISC{Herwig94manyω-categorical, author = {Bernhard Herwig}, title = {Many ω-categorical Structures Have the Small Index Property}, year = {1994} }

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Abstract

Theorem: Let A be a finite Km-free graph, p1,...,pn partial isomorphisms of A. Then there exists a finite extension B, which is also a Km-free graph, and automorphisms fi of B extending the pi’s. This theorem can be used to prove the small index property for the generic countable graph of this class. The same method also works for a certain class of continuum many non isomorphic ω-categorical countable digraphs. Hrushovski proved this theorem for the class of all finite graphs [Hr]; the proof presented here is an extension of his proof. Notation: Let p be a partial mapping on a set A. By D(p) we denote the domain of p by R(p) the range of p (so D(p) ⊂ A and R(p) ⊂ A). In this paper the partial mappings under consideration will always be injective. The edge relation will always be called R; in the first part of the paper we will deal with graphs, so R will be symmetric and irreflexive, in the second part