I will just clarify Shelah's result.  I am not sure that it really corresponds to the original issue. Morley conjecture in the 1960's that the number of models (up to isomorphism) a complete first order theory with cardinal $\kappa$, should be increasing in $\kappa$ (with the one known exception of the theories such as algebraically closed fields which have countably many countable models and one in all larger cardinalities.

Shelah's remarkable strategy for answer this was to conjecture around 1970 all possible spectrum function - a short list but with parameters. Later in the 70's he showed this was 'eventually correct'. Work of Hart, Hrushovski, and Laskowski cleaned up the smaller cardinals.  The idea of the argument is to show that either a theory has a property which implies the maximal number of models or the failure of this conditions takes one toward assigning invariants. (e.g. unstable - there is a formula $\phi(\bar{x},\bar{y})$ and a sequence $\bar{a}_i, \bar{b}_i$ such that $\phi(\bar{a}_i, \bar{b}_j)$ iff $i < j$ - this is called the order property; well order does nor arise here. A collection of 5 such dichomities leads one to classiable theories; every model is determined by a 'cardinal invariant' and so the number of models in $\kappa$ is bounded well-below $2^\kappa$.