Dependent $T$ and existence of limit models

Abstract

Does the class of linear orders have (one of the variants of) the so called $(\lambda,\kappa)$-limit model? It is necessarily unique, and naturally assuming some instances of G.C.H. we get some positive results. More generally, letting $T$ be a complete first order theory and for simplicity assume G.C.H., for regular $\lambda > \kappa > |T|$ does $T$ have (variants of) a $(\lambda,\kappa)$-limit models, except for stable $T$? For some, yes, the theory of dense linear order, for some, no. Moreover, for independent $T$ we get negative results. We deal more with linear orders.