Vaught conjecture for uncountable languages Recall Vaught conjecture:  the number of countable models of a first-order complete theory in a countable language is finite or $\aleph_0$ or $2^{\aleph_0}.$
Now let $\lambda$ be an uncountable cardinal. Is the following version of Vaught conjecture true:
Vaught conjecture for uncountable languages. If $T$ is a complete theory in a language of size $\lambda,$ and if $T$ has more than $\lambda-$many non-isomorphic models of size $\lambda$, then $T$ has $2^\lambda$ many non-isomorphic models models of size $\lambda.$
Also, do we have the following version of Moreley theorem:
Morley theorem for uncountable languages. If $T$ is a complete theory in a language of size $\lambda,$ and if $T$ has more than $\lambda^+-$many non-isomorphic models of size $\lambda$, then $T$ has $2^\lambda$ many non-isomorphic models models of size $\lambda.$
 A: Suppose that $2^{\aleph_{1}} > 2^{\aleph_{0}} > \aleph_{2}$, and consider the language with unary predicates $P_{n}$ $(n \in \omega)$ and $Q_{\alpha}$ $(\alpha < \omega_{1})$. Consider the theory $T$ which says that (1) each $P_{n}$ is satisfied by infinitely many points (2) no point satisfies more than one $P_{n}$ (3) no point satisfies any $Q_{\alpha}$. 
This appears to me to be complete, as for each $n \in \omega$ you can argue by complexity of formulas that for any $k$-ary formula $\phi$ not using any of $P_{m}$ ($m \geq n$), any two models $M$ and $N$ of $T$ and any two tuples $\langle a_{0},\ldots,a_{k-1} \rangle$ from $M$ and $\langle b_{0},\ldots,b_{k-1} \rangle$ from $N$, if 
$M \models P_{p}(a_{i})$ if and only if $N \models P_{p}(b_{i})$ for each $p < n$ and each $i < k$ then $M \models \phi(a_{0},\ldots,a_{k-1})$ if and only if $N \models \phi(b_{0},\ldots,b_{k-1})$. Using this fact, one can show that the second player wins the back-and-forth game using $M$ and $N$ of any fixed finite length, restricting to any finite set of the $P_{n}$'s. 
Now $T$ appears to have exactly continuum many models of cardinality $\aleph_{1}$ up to isomorphism, determined by how many points satisfy each $P_{n}$ (two options in each case), and how many satisfy none of them (countably many options). 
