Extending partial isomorphisms for the small index property of many \(\omega\)-categorical structures
For a finite \(K_{m}\)-free graph \(A\) (\(K_{m}\) is the complete graph with \(m\) vertices) and partial isomorphisms \(p_{1},\ldots, p_{n}\) on \(A\), it is proven that there are a finite extension \(B\), which is also a \(K_{m}\)-free graph, and automorphisms \(f_{i}\) of \(B\), extending the \(p_{i}\) (the property was earlier called the extension property). \textit{W. Hodges, I. Hodkinson, D. Lascar}, and \textit{S. Shelah} [J. Lond. Math. Soc., II. Ser. 48, No. 2, 204-218 (1993; Zbl 0788.03039)] shows that the property is important to understand the automorphism group of a random graph as a topological group. The property was used in the cited paper to prove the small index property (a countable structure \(M\) has the small index property if for every subgroup \(H\) of \(G={\text{Aut}}(M)\), \(H\) is open whenever \([G:H]<2^{\omega}\)) for random graphs. Analogous results are proven here for the generic countable \(K_{m}\)-free graphs and for the generic \(\omega\)-categorical digraphs introduced by \textit{C. W. Henson} [J. Symb. Log. 37, No. 3, 494-500 (1972; Zbl 0259.02040)].
scientific article; zbMATH DE number 1239660
scientific article