Fixed-cardinal control of extension multiplicities

Determine whether, for a complete theory T failing the Gaifman property, the value of I_T(\kappa,\kappa) can be bounded or otherwise controlled by a cardinal fixed independently of the size \kappa of the models.

Background

The quantity I_T(\kappa,\kappa) measures the supremum, over P-parts of size \kappa, of the number of non-isomorphic-over-the-base models of size \kappa extending a fixed P-part. The paper constructs theories failing the Gaifman property for which this quantity can equal values such as \ded(\kappa) or \kappa^{\aleph_0}. It then asks whether this multiplicity can instead be controlled by a fixed cardinal when the theory fails the Gaifman property. The preceding constructions do not resolve the issue because their behavior depends on compactness and on the size of the model.