Vaught's conjecture for first order theories (VC) in its original form states that every complete first order theory of a countable language has either at most countably many isomorphism classes of countable models or else continuum many.
Let $L:=\langle E \rangle$ be a first order language with the binary relation symbol $E$. In the book "The Descriptive Set Theory of Polish Group Actions" by H. Becker and A. Kechris theorem 6.1.3 claims that VC holds if and only if VC holds for $L$-theories of graphs.
The main idea behind this result is that every countable model of a countable language can be interpreted in a graph (seen as a $L$ structure) as demonstrated for example in section 5.5 of Wilfrid Hodges' book "Model Theory".
Let $S$ be an arbitrary countable relational language and $T_1$ a satisfiable $S$-theory. There is a $L$-formula $\phi_D(v_1)$ and a $L$-theory $T_2$ such that for every $L$-model $\mathcal{M}$ of $T_2$ we have $\mathcal{M}\models\exists v_1\phi_D(v_1)$. Furthermore, for each $k<\omega$ and each $k$-ary relation symbol $R$ of $S$ there is a $L$-formula $\psi_R(v_1,\dots ,v_k)$ such that for every model $\mathcal{M}$ of $T_2$ we can define a $S$ model $I(\mathcal{M})$ of $T_1$ in the following way:
1.) The universe of $I(\mathcal{M})$ is $\{a\in M\mid \mathcal{M}\models\phi_D(a)\}$.
2.) A $k$-ary relation symbol $R$ of $S$ is interpreted as $$ \{(a_1,\dots ,a_k)\in I(\mathcal{M})^k\mid \mathcal{M}\models\psi_R(a_1,\dots, a_k)\} $$ If $\mathcal{A}$ is a $S$-model of $T_1$ then we can define a model $\mathcal{M}$ of $T_2$ such that $I(\mathcal{M})\cong\mathcal{A}$.
In case $S$ is finite, VC for $S$-theories reduces to VC for $L$-theories because we can easily choose $T_2$ such that for all models $\mathcal{M}, \mathcal{M'}$ of $T_2$, we have $$\mathcal{M}\cong\mathcal{M'}\Leftrightarrow I(\mathcal{M})\cong I(\mathcal{M'}).$$
My problem is that I cannot define $T_2$ in such a way if the language $S$ is infinite. It can happen that $T_1$ has only countably many isomorphism classes while $T_2$ has continuum many. So my question is how can VC for $S$-theories be reduced to VC for $L$-theories if $S$ is infinite?
A possible approach I have been thinking about is to interpret $T_1$ in some other finite language $L_2$ thereby getting a theory $T_3$ which has the same number of isomorphism classes as $T_1$. Can someone help me with that? Thanks!
By the way, the main idea to solve this problem, for those who are interested -apparently not too many- can be found here:
http://home.mathematik.uni-freiburg.de/ziegler/preprints/INTERPR.pdf