# Theories whose models of cardinality $\kappa$ are all elementary equivalent

A (first-order) theory is $\kappa$-categorical if all its models of cardinality $\kappa$ are isomorphic. Now is there a similar notion which doesn't require isomorphism, but only elementary equivalence? (Let's say "elementary categorical".) Some questions about:

1) Are there any theories with this property (for some $\kappa$ for which they have a model of cardinality $\kappa$) which are not $\kappa$-categorical? And are there any which are not complete?

2) Are there any theories which satisfy the property for every $\kappa$, and are not $\kappa$ categorical for any $\kappa$? And are there any which are not complete?

Sorry for the lack of preciseness in the question, but being a somehow "unheard" problem for me I didn't really know how to state it.

• The theory of algebraically closed field of characteristic 0 is not $\aleph_0$-categorical, but every two model of the theory is elementarily equivalent. The theory of dense linear orders is also good example. It is not $\kappa$-categorical for all uncountable $\kappa$ but the $\aleph_0$-categoricity implies elementary equivalence. Jul 3, 2017 at 15:56

Assuming the language is countable "elementary $\kappa$-categoricity" of $T$ as you define is equivalent to $$T \cup \{\exists x_1, ..., x_n \bigwedge_{i < j} x_i \neq x_j : n < \omega\}$$ being complete. (A contradictory theory is asumed complete here.) In particular this does not depend on $\kappa$.
An example where $T$ is not complete is the theory of vector spaces over a finite field. An example of where $T$ is not categorical is your favourite complete theory that is not categocial, e.g. $DCF_0$.