Applications of Kolmogorov complexity to computable model theory

In this paper we answer the following well-known open question in computable model theory. Does there exist a computable not א0-categorical saturated structure with a unique computable isomorphism type? Our answer is affirmative and uses a construction based on Kolmogorov complexity. With a variation of this construction, we also provide an example of an א1-categorical but not א0-categorical saturated Σ1-structure with a unique computable isomorphism type. In addition, using the construction we give an example of an א1-categorical but not א0-categorical theory whose only non-computable model is the prime one. §