The work presented is joint with S-D Friedman and T Hyttinen. Our aim was to generalize a very nice result of Friedman, Hyttinen, and Rautila which tied first-order model theoretic classification theory to constructibility under the assumption of 0#, to a non-elementary model theoretic setting. The original result stated:

Theorem. Assume 0# exists and let T be a constructible first-orer theory which is countable in the constructible universe L. Let kappa be a cardinal in L larger than (aleph_1)^L. Then the collection of constructible pairs of models A,B of T, |A|,|B|=kappa, which are isomorphic in a cardinal- and real-preserving extension of L is itself constructible if and only if T is classifiable (i.e. superstable with NDOP and NOTOP).

We have chosen Homogeneous Model Theory as a good setting for generalizing this result because of the already well developed structure/non-structure theory in this setting. I will present the result for Homogeneous Model Theory analogous to the theorem above, and discuss some of the issues involved in the proof.