Abstract
We consider the question on diagram preserving extension of $$ D $$ -types. Let $$ D $$ denote the diagram of a $$ \lambda $$ -homogeneous model and let $$ S $$ be a set of $$ D $$ -types whose domains are subsets of cardinality less than $$ \lambda $$ of a suitable $$ D $$ -set $$ B $$ . We prove that a model $$ V_0 $$ 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 $$ V_0 $$ does not exceed $$ \lambda $$ and $$ \lambda =\kappa ^+ $$ then it is preserved in $$ V $$ . We also find conditions for preservation of all cardinals of $$ V_0 $$ in $$ V $$ .