Consider the language $\mathcal{L}$ of first-order logic equipped with a single binary relation symbol $R$. Let $\Gamma$ denote the theory expressing that $R$ is an equivalence relation such that there is exactly one equivalence class of size $n$ for each $n\in \mathbb{Z}_{>0}$.
Is it the case that $\Gamma$ is $\kappa$-categorical for every
uncountable cardinal $\kappa$?
I've been unable to make progress in determining the answer. Any help would be appreciated!
Best Answer
This theory is complete, but not $\kappa$-categorical. The conditions don't say anything about the number of infinite equivalence classes, so you can construct two models, one with a single infinite equivalence class, the other with $\kappa$-many such classes. These are clearly not isomorphic, so the theory is not categorical.
In fact, for every infinite cardinal $\kappa$ and every model M of cardinality $\leq \kappa$ we can find an elementary extension $N_\kappa$ of $M$ with $\kappa$ infinite equivalence classes and with each infinite equivalence class having cardinality $\kappa$. This extension is unique up to isomorphism for each cardinal $\kappa$, so the theory is complete.