$\omega$-categoricity and infinite languages The Ryll-Nardzewski Theorem states that an equivalent condition to $\omega$-categoricity is that there is a finite number of $n$-types for any $n$.
So what happens when you add a countably many unary predicate to the signature of an $\omega$-categorical theory?
Two examples: 


*

*Let $\mathcal{G}_1$ be infinite empty graph, augmented with predicates $P_i$ each true on $i$ vertices. 

*Take an enumeration of rigid graphs $(G_i)$ of increasing size, such that $P_i$ is true only on the vertices of $G_i$. Let $\mathcal{G}_2$ to be the union of these graphs.


How many models do the theory of these graphs have? Can they all obtained by some standard model-theoretic construction? 
Thanks for your help.
 A: *

*Let $M$ be our $\omega$-categorical structure, and let $\hat{M}$ be $M$ expanded by the predicates $\{P_i\mid i\in\omega\}$ and say that two predicates $P_i$ and $P_j$ are equivalent if they have the same interpretation on $M$. As you note, Ryll-Nardzewski tells us that in an $\omega$-categorical theory, there are only finitely many $1$-types, so $\text{Th}(\hat{M})$ has no hope of being $\omega$-categorical, unless the new predicates are partitioned into only finitely many equivalence classes (i.e. if we only really added finitely many predicates). But even if we only add $1$ predicate, we can still get a theory which isn't $\omega$-categorical, see point 4.


*Here's an example where we easily get $2^{\aleph_0}$-many countable models. Let $M$ be an infinite set (a structure in the empty language), and interpret the $P_i$ so that for any finite sets $X,Y\subset \omega$ such that $X\cap Y = \emptyset$, there exists an $x$ such that $\hat{M}\models P_i(x)$ for $i\in X$ and $\hat{M}\models \lnot P_i(x)$ for $i\in Y$. Then there are already continuum-many quantifier-free $1$-types consistent with $\text{Th}(\hat{M})$, so this theory has continuum-many countable models.


*You can also get exactly $n$ models for any $n\in \omega$, $n \neq 2$ as an expansion of DLO with countably many unary predicates. Take the standard examples and replace the constant symbols with predicates that pick out exactly one element.


*It's a fun fact (see Section 5.5 of Hodges' big Model Theory) that for any first-order structure $N$ in a finite language $L$, there is a graph $G$ such that $N$ and $G$ are bi-interpretable. Given a countable structure $N$, let $G$ be the (countable) graph which interprets it, and let $M$ be the random graph. Then $G$ embeds in $M$, since $M$ is universal for countable graphs. Let $P$ be a new predicate and expand $M$ to $\hat{M}$ by letting $P$ pick out the copy of $G$ in $M$. Then $\text{Th}(\hat{M})$ interprets $\text{Th}(G)$, which interprets $\text{Th}(N)$. So, for example, by adding a single predicate to the random graph we can obtain a theory which interprets ZFC set theory.
