Extending Types Preserving Diagrams

We consider the question on diagram preserving extension of D -types. Let D denote the diagram of a λ -homogeneous model and let S be a set of D -types whose domains are subsets of cardinality less than λ 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 λ and λ =κ ⁺ then it is preserved in V . We also find conditions for preservation of all cardinals of V₀ in V .