# Adding unary relation symbol within complete theory

I try to prove following problem: Let $T$ be a complete theory over countable language, then $T$ has a model $\mathfrak{A}$ with cardinality $\le 2^{\aleph_0}$ such that for each $\mathfrak{B}\models T$ and $S\subset \mathfrak{B}$ there is $R\subset \mathfrak{A}$ such that $(\mathfrak{B},S)$ is elementary equivalent to $(\mathfrak{A},R)$.

By Löwenheim–Skolem theorem, we can only consider countable models of $T$ and its subsets, but I don't know how to proceed to prove it. Any hint would be appreciated.

• You may look at the notion of resplendent structure. – Primo Petri Sep 15 '14 at 6:58

Some hints. First use Lowenheim Skolem to show that is suffices to consider countable $(\mathfrak{B}, S)$. Then take $\mathfrak{A}$ to be countably saturated.
• @tetori: This is almost the definition of saturation. Try constructing $\mathfrak A$ recursively. – tomasz Sep 15 '14 at 20:05