From the text: An important open problem in the field of computable models is the Goncharov-Ash problem about the relation between the algorithmic dimension of a model \({\mathcal B}\) and of its enrichment \(({\mathcal B},c_1,c_2,\dots, c_m)\) by a finite set of constants. Note that the inequality \[ \dim({\mathcal B})\leq \dim({\mathcal B},c_1,c_2,\dots, c_m) \] is always fulfilled.
The following theorem gives a complete solution to the Goncharov-Ash problem.
Theorem 1. For any nonzero cardinal \(n\leq\omega\), there exists an autostable model \({\mathcal B}\) such that the algorithmic dimension of the model \(({\mathcal B},c)\) equals \(n\) for an arbitrary constant \(c\in{\mathcal B}\).