Is this theory $\kappa$-categorical when $\kappa$ is infinite and not $\le \aleph_0$? Let $\mathcal L$ be a language with only a binary predicate $E$, and let $T$ be a theory of structures in which $E$ is an equivalence relation which partitions the structure into two infinite sections. It is easy to see that $T$ is $\aleph_0$-categorical but not $\aleph_\alpha$-categorical when $\alpha>0$.
My question: can there be a $\kappa \not \le \aleph_0$ such that $T$ has a model of cardinality $\kappa$ and is $\kappa$-categorical? What about $\kappa$ is infinite but Dedekind-finite?
 A: If $M$ is a model of $T$ let $M_0$ and $M_1$ denote the two parts defined by $E$.
Suppose that $A$ is an infinite Dedekind-finite set. If $A$ is amorphous then it cannot be a model of $T$ to begin with because it cannot be split into two infinite sets.
Suppose that $A$ can be split into $A_0$ and $A_1$. If $|A_0|=|A_1|+\frak a$ for $\frak a$ nonzero, then taking any $B\subseteq A_0$ such that $0<|B|<\frak a$ defines a non-isomorphic model by $A'_0=A_0\setminus B$ and $A'_1=A_1\cup B$. To see that this partition is not isomorphic to the previous partition note that it could only happen if $|A'_1|=|A_0|$ (because we remove points from $A_0$ and its cardinality is strictly smaller now) but this means there is a bijection $f\colon A_0\to A_1\cup B$ which is false because $|A_0|=|A_1|+|B|=|A_1|+\frak a$ and therefore $|B|=\frak a$, in contradiction to the way we chose $B$.
If $A$ can be split into two incomparable parts, then any exchange of points will keep it incomparable by a similar argument.

However it is possible to get categoricity in $\aleph_1$-amorphous cardinals. Namely cardinals of sets that cannot be split into two disjoint uncountable sets.
Suppose that $A$ is $\aleph_1$-amorphous, and it has no infinite Dedekind-finite subset, then any partition of $A$ into two parts would have to have one part of size $\aleph_0$ and another part of size $|A|$. Therefore categoricity follows for any $\aleph_1$-amorphous cardinal.

Examining this proof shows that $T$ is categorical in $\kappa$ if and only whenever $\kappa=\frak a+b$ then we can guarantee that either $\frak a$ or $\frak b$ is $\aleph_0$, and the other is Dedekind-infinite.
This happens if and only if $A$ cannot be split into two uncountable sets, has no infinite Dedekind-finite subsets, and if $B\subseteq A$ is infinite then $|B|=|A|$ or $|B|=\aleph_0$.
