Symmetry for non-μ-splitting over limit models from μ-superstability alone, and the equivalence of uniqueness of limit models with superstability in abstract elementary classes

In an abstract elementary class (AEC) with amalgamation satisfying the local-character form of μ-superstability (VanDieren's Assumption 1), it is open whether symmetry for non-μ-splitting over limit models must hold, equivalently whether every reduced tower of limit models is continuous, when no locality (tameness), good-frame, or saturated-union hypothesis is assumed. A positive answer would establish that μ-superstability alone implies uniqueness of limit models of cardinality μ — the superstability-to-uniqueness direction of a long-standing conjecture — without tameness; the converse direction is known only under tameness. The question was isolated by VanDieren (2016), left explicit as Question 4.13 of VanDieren–Vasey (2017), and remains the standing missing input in the 2024–2026 literature on limit models; no counterexample is known.

Background

An abstract elementary class (AEC) is an abstract class of structures $\mathcal{K}$ with a strong substructure relation $\prec_{\mathcal{K}}$, closed under isomorphism and directed colimits, with a Löwenheim–Skolem number $LS(\mathcal{K})$ (every structure in the class has a $\prec_{\mathcal{K}}$-closed substructure of size at most $LS(\mathcal{K})$); it generalizes the class of models of a complete first-order theory with elementary embedding, but without a syntax. Types are semantic: the Galois type $ga\text{-}tp(a/M)$ of an element $a$ over $M \in \mathcal{K}$ is its orbit under automorphisms of a sufficiently large homogeneous 'monster' model fixing $M$. A type $p = ga\text{-}tp(a/M)$ μ-splits over $N \prec_{\mathcal{K}} M$ if there exist $N_1, N_2 \prec_{\mathcal{K}} M$ with $N \prec_{\mathcal{K}} N_1$ and $N \prec_{\mathcal{K}} N_2$ and an isomorphism $h: N_1 \cong N_2$ with $h \upharpoonright N = \mathrm{id}_N$ such that $h(p \upharpoonright N_1) \neq p \upharpoonright N_2$; $p$ does not μ-split over $N$ when no such $N_1, N_2$, and $h$ exist. Non-splitting is the basic forking-like independence notion available without syntax.

Fix $\mu \geq LS(\mathcal{K})$ and assume joint embedding (any two models of the class embed into a common one), amalgamation (any two embeddings of a common model into models of the class can be completed to a commuting triangle inside the class), and no maximal models of cardinality $\mu^+$ (every model of cardinality $\mu^+$ has a proper $\prec_{\mathcal{K}}$-extension). $M \in \mathcal{K}_\mu$ is a $(\mu,\theta)$-limit model over $M_0$ ($\theta < \mu^+$ limit) if $M = \bigcup_{i<\theta} M_i$ with each $M_{i+1}$ universal over $M_i$ (every $N \in \mathcal{K}_\mu$ containing $M_i$ embeds into $M_{i+1}$ over $M_i$). Uniqueness of limit models — any two limit models over the same base, of possibly different lengths $\theta_1, \theta_2 < \mu^+$, are isomorphic over it — is a central structural goal: it underlies instances of Shelah's categoricity conjecture, construction of saturated models in singular cardinals, and downward categoricity transfer.

VanDieren's Assumption 1 is a local-character formulation of μ-superstability: (1) $\mathcal{K}$ is Galois-stable in $\mu$ (every $M \in \mathcal{K}_\mu$ has at most $\mu$ Galois types over it); (2) μ-splitting satisfies locality/continuity (a type all of whose restrictions along a chain do not μ-split over $N$ does not μ-split over $N$) and there are no long splitting chains (along any increasing chain of limit models of length $\alpha$, every type over the union stops μ-splitting over some $M_i$). Under Assumption 1 plus the additional hypothesis that unions of chains of $\mu^+$-saturated models are $\mu^+$-saturated, VanDieren proved that reduced towers are continuous (Theorem 2) and consequently that $(\mu,\theta)$-limit models are unique (Theorem 1). A tower is a triple $(\bar{M}, \bar{a}, \bar{N})$: an increasing sequence $\bar{M} = \langle M_i : i < \alpha \rangle$ of limit models with distinguished elements $a_i \in M_{i+1} \setminus M_i$ whose types over $M_i$ do not μ-split over $N_i \prec M_i$, with $M_i$ universal over $N_i$; it is reduced if for every tower extension $(\bar{M}', \bar{a}, \bar{N})$ of $(\bar{M}, \bar{a}, \bar{N})$ and every $i < \alpha$, $M'_i \cap \bigcup_{j<\alpha} M_j = M_i$ (VanDieren 2016), and continuous if $M_\beta = \bigcup_{j<\beta} M_j$ at every limit $\beta$. Crucially, VanDieren's Theorem 3 shows that under Assumption 1 alone, continuity of all reduced towers is equivalent to symmetry for non-μ-splitting over limit models (Definition 3): whenever $M$ is universal over the limit model $M_0$, $M_0$ limit over $N$, $ga\text{-}tp(a/M_0)$ is non-algebraic (having infinitely many possible realizations) and does not μ-split over $N$, and $ga\text{-}tp(b/M)$ is non-algebraic and does not μ-split over $M_0$, there is a limit model $M_b$ over $M_0$ containing $b$ with $ga\text{-}tp(a/M_b)$ not μ-splitting over $N$.

All known routes to symmetry import hypotheses beyond Assumption 1: μ-tameness (any two distinct Galois types over a model of size $\mu$ already differ over a small submodel — a locality property for types), the existence of a good $\mu^+$-frame (Shelah's axiomatic forking calculus, which builds symmetry in by fiat at $\mu^+$), saturated unions at $\mu^+$, or categoricity with failure of the order property (a stability-theoretic condition preventing the class from defining an infinite linear order). Grossberg and Vasey proved that the various 'shades' of superstability — local character of splitting, uniqueness of limit models, no long splitting chains — are equivalent under tameness; without tameness the dividing line remains uncharted, and VanDieren's series of papers explicitly targets the still-open conjecture that uniqueness of limit models of cardinality $\mu$ is equivalent to superstability.

Problem Statement

Let $\mathcal{K}$ be an abstract elementary class with the joint embedding and amalgamation properties and no maximal models of cardinality $\mu^+$, satisfying Assumption 1 (μ-superstability: Galois-stability in $\mu$, locality/continuity of μ-splitting, and no long splitting chains over limit models). Determine, without assuming tameness or any other locality property for types, without assuming the existence of a good $\mu^+$-frame, and without assuming that unions of chains of $\mu^+$-saturated models are $\mu^+$-saturated:

1. Does $\mathcal{K}$ necessarily have symmetry for non-μ-splitting over limit models — in the sense of VanDieren's Definition 3: if $M$ is universal over the limit model $M_0$, $M_0$ is a limit model over $N$, $ga\text{-}tp(a/M_0)$ is non-algebraic and does not μ-split over $N$, and $ga\text{-}tp(b/M)$ is non-algebraic and does not μ-split over $M_0$, then there exists a limit model $M_b$ over $M_0$ containing $b$ such that $ga\text{-}tp(a/M_b)$ does not μ-split over $N$? By VanDieren's Theorem 3 this is equivalent to: is every reduced tower of limit models of cardinality $\mu$ continuous?
2. Consequently, does μ-superstability alone imply that any two limit models over the same base are isomorphic over it — a $(\mu,\theta_1)$-limit model and a $(\mu,\theta_2)$-limit model, for limit ordinals $\theta_1, \theta_2 < \mu^+$ possibly of different lengths — and is uniqueness of limit models of cardinality $\mu$ in fact equivalent to superstability (the converse direction being known, so far, only under tameness)?

Item 1 admits either resolution: a proof that symmetry (equivalently, by VanDieren's Theorem 3, continuity of all reduced towers of limit models of cardinality $\mu$) holds under Assumption 1 alone, or an explicit abstract elementary class satisfying the hypotheses in which symmetry fails, i.e. containing a discontinuous reduced tower. Uniqueness of limit models is a consequence of a positive resolution rather than an equivalent formulation of it: continuity of reduced towers suffices for uniqueness of limit models, while the converse implication is not known. A proof that μ-superstability implies uniqueness of limit models which bypassed symmetry and reduced-tower continuity would therefore be progress on item 2 but would not by itself resolve item 1. Item 2 is a deliverable in its own right and is not discharged by either resolution of item 1: a proof of symmetry settles only its first clause (μ-superstability implies uniqueness of limit models), and a discontinuous reduced tower does not by itself settle either clause, since failure of symmetry does not refute uniqueness of limit models. Its equivalence clause — uniqueness of limit models of cardinality $\mu$ equivalent to superstability — requires in addition the remaining direction, uniqueness of limit models implies superstability, the open converse known so far only under tameness, to be proved or refuted separately.

References

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