The concept that I want to think about in this question is very similar to trace definability, introduced here and also independently by Guingona, but is somewhat different from it. There's a very good chance that this weak trace interpretability has a name in the literature, but I have been unable to find it.
Take all theories to be countable. Consider the following restatements of standard facts:
A theory $T$ is unstable if and only if there exists a model $M \models T$, an injection $f: \mathbb{Q} \to M^n$ for some $n$, and a formula $\varphi(\bar{x},\bar{y})$ on $2n$-tuples such that for any $a,b \in \mathbb{Q}$, $a<b$ if and only if $M \models \varphi(f(a),f(b))$.
A theory $T$ is un-$\omega$-stable if and only if there exists a model $M \models T$, an injection $f: 2^\omega \to M^n$ for some $n$, and a family of formulas $\{\varphi_\sigma(\bar{x}) : \sigma \in 2^{<\omega}\}$ on $n$-tuples such that for every $\alpha \in 2^\omega$ and every $\sigma \in 2^{<\omega}$, $\sigma$ is an initial segment of $\alpha$ if and only if $M \models \varphi_{\sigma}(f(\alpha))$.
So given an $\mathcal{L}$-structure $N$ with $\mathcal{L}$ a relational language, we'll say that a structure $M$ weakly trace interprets $N$ if there is an injection $f: N \to M^n$ for some $n$, and a formula $\varphi_P(\bar{x}_1,\dots,\bar{x}_k)$ on $kn$-tuples for each $k$-ary predicate symbol $P\in \mathcal{L}$ such that for any $k$-ary $P \in \mathcal{L}$ and any $k$-tuple $\bar{a} \in N^k$, $N \models P(\bar{a})$ if and only if $M \models \varphi_P(f(a_1),\dots,f(a_k))$. (The difference between this and trace definability is that trace definability requires that all definable relations in $N$ be represented in $M$.) We'll say that a theory $T$ weakly trace interprets a theory $S$ if some model of $T$ weakly trace interprets some model of $S$.
So now we can succinctly say that a theory is stable if and only if it does not weakly trace interpret $(\mathbb{Q},<)$ and is $\omega$-stable if and only if it does not weakly trace interpret $2^{\omega}$ with predicates $U_\sigma$ each selecting out the set of elements with initial segment $\sigma$. A result in Walsberg's paper also more or less states that a theory is NIP if and only if it does not weakly trace interpret the random graph. (Since all three of these structures admit quantifier elimination, the notion of weak trace interpretation I am discussing here is equivalent to trace definability.)
Trace definability is better behaved in terms of type counting, and in particular Walsberg shows that if $T$ is a $\kappa$-stable theory and $S$ is trace definable in $T$, then $S$ is a $\kappa$-stable theory. I'm curious if this result can be extended to this notion of weak trace interpretation, and I'm wondering if there is a single canonical witnessing structure for unsuperstability.
| Question 1. If $T$ is a superstable theory that weakly trace interprets $S$, does it follow that $S$ is superstable? (Equivalently, is there a superstable theory that weakly trace interprets a strictly stable theory?)
| Question 2. Is there a single structure $M$ such that $T$ is superstable if and only if $T$ does not weakly trace interpret $M$?
I vaguely expect the answer to the second question to be no, but I don't see an approach to proving it.
EDIT: Erik Walsberg has pointed out in the comments that Question 1 has a negative answer.
