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 Ho-categorical saturated structure with a unique computable isomor phism 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 H i-categorical but not Hq-categorical saturated D?structure with a unique computable isomorphism type. In addition, using the construction we give an example of an K?-categorical but not Ho-categorical theory whose only non-computable model