I am trying to figure out whether the following structure is $\omega$-categorical.
The language contains countably many binary relations $E_n$ and a binary relation $<$.
The structure itself is a countable tree where each element has infinitely many children, the tree order is the interpretation of $<$.
The relation $E_i$ is a partial equivalence class whose domain is the $i$th level of the tree. It has the property that for each element $a$ on the $(i-1)$th level there are infinitely many of the children of $i$ in any equivalence class.
In a countably categorical theory $T$, for every $n$, there are only finitely many formulas in $n$ variables up to equivalence modulo $T$. Your language already includes infinitely many binary relations which are not equivalent in models of $T$, so $T$ is obviously not countably categorical.