Does Lascar rank at least $\omega^\omega$ force continuum many countable models? (Tanović's conjecture for superstable theories)

For a complete countable superstable theory $T$ , let $U(T)$ be the supremum of the Lascar $U$ -ranks of its complete types and $I(T,\aleph_0)$ the number of its non-isomorphic countable models. Tanović (2007) conjectured that $U(T)\ge\omega^\omega$ implies $I(T,\aleph_0)=2^{\aleph_0}$ ; he proved it for trivial and for one-based theories, and in 2013 reduced the general case to the assertion that the generic types of the simple groups definable in $T^{eq}$ of $U$ -rank below $\omega^\omega$ are eventually strongly non-isolated. This record states the problem and its literature status: the conjecture remains open — no proof or counterexample is known, even in the $\aleph_0$ -stable special case — as confirmed by the Pillay–Tanović survey of August 2026.

Background

For a complete first-order theory $T$ in a countable language, $I(T,\aleph_0)$ denotes the number of non-isomorphic countable models of $T$ , always at most $2^{\aleph_0}$ . Vaught's conjecture (1961) predicts that $I(T,\aleph_0)$ is never strictly between $\aleph_0$ and the continuum: every countable complete theory has either countably many or exactly $2^{\aleph_0}$ countable models. It is one of the oldest open problems of model theory, and the stability hierarchy is its most developed testing ground.

A theory is stable if over parameter sets of some sufficiently large infinite cardinality it has no more complete 1-types than the size of the parameter set; it is superstable if this holds over all sufficiently large sets, and $\aleph_0$ -stable (equivalently $\omega$ -stable) if it holds already over countable sets. Superstable theories carry Lascar's $U$ -rank, an ordinal-valued rank measuring iterated forking, where forking is stability theory's calibrated notion of algebraic dependence: informally, $U(p)$ is at least $\alpha+1$ exactly when the type $p$ has a forking extension of rank at least $\alpha$ . The rank of the theory, $U(T)$ , is the supremum of the ranks of complete types over the empty set, so $U(T)\ge\omega^\omega$ means that for every natural number $n$ the theory has types of rank at least $\omega^n$ — the genuinely infinite-rank regime, beyond the finite-rank theories handled by classical structure theorems.

Isolation of types drives the counting of countable models. A type over a finite set is isolated if a single formula already implies it, and eventually non-isolated (ENI) if some non-forking extension over a larger finite set is non-isolated. For $\omega$ -stable theories, Shelah, Harrington and Makkai proved Vaught's conjecture by showing that having fewer than $2^{\aleph_0}$ countable models forces the tree of ENI strongly regular types to have depth at most 1, from which countable models are classified by dimensions and number at most $\aleph_0$ . Here a strongly regular type is a refinement of the notion of a regular type (a non-algebraic type all of whose forking extensions are orthogonal to it) which, like a strongly minimal set (an infinite definable set with no infinite proper definable subset), supports a single well-behaved dimension; two types are orthogonal when no non-forking extensions of them over a common parameter set share a dimension. Buechler later proved Vaught's conjecture for superstable theories of finite $U$ -rank, and Low and Pillay showed that a superstable, non- $\omega$ -stable theory with few countable models must contain an infinite definable group whose generic types are regular and locally modular (a tameness condition on the associated forking pregeometry).

For superstable theories of arbitrary rank the question is open. Tanović's 2007 paper introduced stronger, domain-stable isolation notions: a type is strongly non-isolated if the complete types almost orthogonal to it (orthogonal over their common finite domain) are dense in every finite-variable type space over every finite extension of its domain, and eventually strongly non-isolated (ESN) if some non-forking extension is strongly non-isolated. He proved a dichotomy for countable superstable or small stable theories (small meaning only countably many complete types over $\emptyset$ ): a regular type over a finite domain is ESN exactly when it is orthogonal to every NENI type (strongly regular, not eventually non-isolated) over a finite domain. Combined with a construction that encodes arbitrary subsets of $\omega$ into pairwise non-isomorphic countable models using an orthogonal family of ESN types of ranks $\omega^n$ , this yielded: if $T$ is superstable and trivial (pairwise independent triples of realizations of a regular type are jointly independent, so the forking pregeometry has no linear or projective structure) or one-based (the canonical base of every stationary type — the parameter data controlling its forking — is algebraic over each of its realizations, a global local-modularity condition), and $U(T)\ge\omega^\omega$ , then $I(T,\aleph_0)=2^{\aleph_0}$ . The paper conjectures the same conclusion for all superstable theories.

In a 2013 follow-up, Tanović reduced the remaining cases to a question about definable groups: the conjecture follows if the generic type of every simple group definable in $T^{eq}$ of rank below $\omega^\omega$ is ESN, and this hypothesis is already verified for superstable fields, so any counterexample must contain infinite simple groups of unboundedly high rank interpreting no field of comparable rank. (Here $T^{eq}$ is the canonical expansion of $T$ by definable quotients, and the generic type of a definable group is its type of maximal rank, whose realizations cover the group up to boundedly many translates.) The August 2026 survey by Pillay and Tanović records Vaught's conjecture for superstable theories as open in full generality and lists the enabling statement — the generic type of any simple superstable group is eventually non-isolated — among the area's open problems. Notably, the conjecture is open even for $\aleph_0$ -stable theories, where the Vaught dichotomy itself is a theorem: the ENI-depth analysis gives no bound on Lascar rank.

Problem Statement

Let $T$ be a complete first-order theory in a countable language, with infinite models, and suppose that $T$ is superstable. Let

U(T)=\sup\{\,U(p)\;:\;p\text{ is a complete type over }\emptyset\,\}

denote the Lascar rank of the theory (equivalently, $U(T)\ge\omega^\omega$ holds exactly when for every $n<\omega$ there is a complete type of $U$ -rank at least $\omega^n$ ), and let $I(T,\aleph_0)$ denote the number of non-isomorphic countable models of $T$ .

Determine the truth of Tanović's conjecture: does $U(T)\ge\omega^\omega$ imply $I(T,\aleph_0)=2^{\aleph_0}$ ? Equivalently, in contrapositive form: must a superstable theory with fewer than $2^{\aleph_0}$ non-isomorphic countable models have Lascar rank $U(T)<\omega^\omega$ ?

The implication must be settled at this level of generality, for arbitrary countable superstable theories, with no additional hypotheses such as one-basedness, triviality, smallness, finite rank, $\aleph_0$ -stability, or the absence of definable groups. Currently the conjecture is known only for trivial and one-based theories, and by Tanović's 2013 reduction it would follow from the eventual strong non-isolation of the generic types of all simple groups definable in $T^{eq}$ of $U$ -rank below $\omega^\omega$ ; it is open even for $\aleph_0$ -stable theories. Either a proof or an explicit counterexample (a countable superstable theory with $U(T)\ge\omega^\omega$ and $I(T,\aleph_0)<2^{\aleph_0}$ ) resolves the problem.

References

Predrag Tanović, "Non-isolated types in stable theories", Annals of Pure and Applied Logic 145(1):1–15, 2007. DOI: 10.1016/j.apal.2006.05.014, https://doi.org/10.1016/j.apal.2006.05.014. (Source paper; the conjecture is stated in Section 5.)
Predrag Tanović, "Simple groups and the number of countable models", Archive for Mathematical Logic 52(7–8):779–791, 2013. DOI: 10.1007/s00153-013-0343-x, arXiv:1304.7099, https://doi.org/10.1007/s00153-013-0343-x. (Theorem 1 reduces the conjecture to ESN generic types of simple groups; quotes Conjecture 1 from the 2007 paper and notes it is open even for ℵ0-stable theories.)
Anand Pillay, Predrag Tanović, "The number of countable models of first-order theories", survey preprint, arXiv:2508.06854 (v2, 15 August 2026), https://arxiv.org/abs/2508.06854. (Confirms the superstable case of Vaught's conjecture is open; poses Problem 1 on generic types of simple superstable groups.)
Steven Buechler, "Vaught's conjecture for superstable theories of finite rank", Annals of Pure and Applied Logic 155(3):135–172, 2008. DOI: 10.1016/j.apal.2008.03.004, https://doi.org/10.1016/j.apal.2008.03.004.
S. Shelah, L. Harrington, M. Makkai, "A proof of Vaught's conjecture for ω-stable theories", Israel Journal of Mathematics 49:259–280, 1984. DOI: 10.1007/bf02760651, https://doi.org/10.1007/bf02760651.
Lee Fong Low, Anand Pillay, "Superstable theories with few countable models", Archive for Mathematical Logic 31:457–465, 1992. DOI: 10.1007/bf01277487, https://doi.org/10.1007/bf01277487.
R. L. Vaught, "Denumerable models of complete theories", in: Infinitistic Methods (Proceedings of the Symposium on Foundations of Mathematics, Warsaw 1959), Pergamon Press, Oxford/Warsaw, 1961, pp. 303–321; reprinted in Mathematical Logic in the 20th Century, World Scientific, 2003, pp. 609–627 (reprint pagination). DOI: 10.1142/9789812564894_0028, https://doi.org/10.1142/9789812564894_0028.