Geodesic
Computable graphs of finite $\Delta_\alpha^0$-dimensions
Sibirskij žurnal čistoj i prikladnoj matematiki, Tome 7 (2007) no. 1, pp. 102-113 Cet article a éte moissonné depuis la source Math-Net.Ru

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In present article, we prove the following assertions: For every computable successor ordinal $\alpha$, there exists a $\Delta_\alpha^0$-categorical directed graph (symmetric, irreflexive graph) which is not relatively $\Delta_\alpha^0$-categorical, i.e. no formally $\Sigma_\alpha^0$-Scott family exists for such a structure. For every computable successor ordinal $\alpha$, there exists an intrinsically $\Sigma_\alpha^0$-relation on universe of a computable directed graph (symmetric, irreflexive graph which is not a relatively intrinsically $\Sigma_\alpha^0$-relation. For every computable successor ordinal $\alpha$ and finite $n$, there exists a $\Delta_\alpha^0$-categorical directed graph (symmetric, irreflexive graph) whose $\Delta_\alpha^0$-dimension is equal to $n$. For every computable successor ordinal $\alpha$, there exists a directed graph (symmetric, irreflexive graph) possesing presentations only in the degrees of sets $X$ such that $\Delta_\alpha^0(X)\ne\Delta_\alpha^0$. In particular, for each finite $n$, there exist is a structure with presentations in just the non-low $n$ degrees. 
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J. A. Tussupov. Computable graphs of finite $\Delta_\alpha^0$-dimensions. Sibirskij žurnal čistoj i prikladnoj matematiki, Tome 7 (2007) no. 1, pp. 102-113. http://geodesic.mathdoc.fr/item/VNGU_2007_7_1_a7/

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