Potential isomorphism of elementary substructures of a strictly stable homogeneous model

This results herein are part of a larger project to characterize the classification properties of the class of submodels of a homogeneous stable diagram in terms of the solvability of the potential isomorphism problem for this class of submodels. We assume we work with a large strictly stable homogeneous monster model ç, and that ( = cf( ) > r(ç)). Let caP( r(ç))PIPç be the collection of pairs (A ,B) ∈ L of locally Fç r(ç)-saturated elementary substructures ofç with universe such that there is a cardinaland P( r(ç))-preserving extension of L in which A ∼= B. We show that caP( r(ç))PIPç is equiconstructible with 0 . The proof uses a novel method that does away with the need for a linear order on the