We consider the question on diagram preserving extension of D -types. Let D denote the diagram of a -homogeneous model and let S be a set of D -types whose domains are subsets of cardinality less than of a suitable D -set B. We prove that a model V₀ of the set theory ZFC admits a generic extension V such that each type in S extends to a D -type over B ; moreover, if a cardinal of V₀ does not exceed and = ^+ then it is preserved in V. We also find conditions for preservation of all cardinals of V₀ in V.
K. Zh. Kudaĭbergenov (Sun,) studied this question.