I'm trying to prove Lindström's theorem, which says that a countable inductive first order theory $T$ which is $\kappa$-categorical for some $\kappa\geq \aleph_{0}$ is model-complete.
The strategy I follow is proving each model has an existentially-closed extension of any bigger cardinality, which I proved using transfinite induction. Now, I try to prove an analogous result for non-e.c models - if $T$ has non-e.c. model, then it has a such a model of any infinite size. Using downwards Löwenheim-skolem, we can assume that $M\models T$ is a countable model which is not e.c.
So there is another model $N$ which thinks a certain existential sentence with parameters in $M$ that $M$ does not think. I tried to elementarily extend both $M$ and $N$ in a manner that preserves the inclusion in each step, but it doesn't seem to work.
Once I know this two facts, we obtain 2 models of cardinality $\kappa$ - one of which is e.c. and the other is not, contradicting the fact they are isomorphic. I'd like to know how to fill in this specific part.
Consider the two-sorted structure $(M, N)$ with constants representing elements in $M$ and an added function representing the inclusion map $M \to N$. Let $T$ be the theory of this structure. Add $\kappa$ many constants of the first sort to the language and add axioms stating these constants take distinct values. By compactness, we conclude that this new theory admits a model $(M’, N’)$. Clearly $|M’| \geq \kappa$. Using downward Löwenheim-Skolem if necessary, we may assume $M’$ and $N’$ are both of cardinality exactly $\kappa$. Clearly $M’$ elementarily extend $M$ and $N’ \supset M’$. $N’$ is also elementarily equivalent to $N$ with parameters in $M$. Thus, these structures satisfy the properties you want.