Symmetry, superlimits, and tameness from superstability-like hypotheses in abstract elementary classes with amalgamation

In the classification theory of abstract elementary classes (AECs), the paper 'Symmetry in abstract elementary classes with amalgamation' (VanDieren–Vasey, 2017) proved that $\mu$-superstability together with $\mu$-tameness implies the symmetry property for non-$\mu$-splitting, and recorded precisely what remains unresolved. The problem asks for the exact implication structure between the hypothesis side of the local superstable theory (existence of a type-full categorical good $\mu$-frame, $\mu$-superstability, categoricity in a high-enough cardinal) and its structural-conclusion side (symmetry of splitting and its variants, existence of a superlimit model of size $\mu$, tameness). Its four components are: whether $\mu$-superstability or a type-full categorical good $\mu$-frame implies $\mu$-symmetry (Question 4.13 of the source); which implications among the four variants of $\mu$-symmetry can be reversed and whether they all follow from superstability; whether $\mu$-superstability — or $\mu$-tameness plus a type-full good $\mu$-frame — yields a superlimit model of size $\mu$, equivalently whether unions of chains of limit models of length below $\mu^+$ are limit models (Question 6.12); and whether categoricity in a high-enough cardinal implies full tameness rather than only weak tameness (Grossberg–VanDieren Conjecture 1.5). Later literature resolved important special cases — tameness from categoricity under the weak generalized continuum hypothesis WGCH ($2^\lambda < 2^{\lambda^+}$ for every infinite cardinal $\lambda$; Vasey 2019), symmetry under tameness combined with continuity of nonsplitting (Leung 2024), superlimits on tails of cardinals under tameness (Grossberg–Vasey 2017) — but all four questions remain open in their stated generality, and no separating counterexamples are known.

Background

An abstract elementary class (AEC) is a class $\mathcal{K}$ of structures in a fixed vocabulary together with a strong-substructure relation $\leq_{\mathcal{K}}$ abstracting elementary substructure: the class is closed under isomorphisms and under unions of $\leq_{\mathcal{K}}$-directed systems, satisfies a Löwenheim–Skolem axiom with a cardinal $\operatorname{LS}(\mathcal{K})$, and $\leq_{\mathcal{K}}$ obeys a coherence condition. AECs, introduced by Saharon Shelah, generalize elementary classes and are the standard setting for classification theory beyond first-order logic, with Shelah's categoricity conjecture as the main test question. Throughout, $\mathcal{K}_\mu$ denotes the models of cardinality $\mu \ge \operatorname{LS}(\mathcal{K})$, and the source work assumes amalgamation (Hypothesis 2.1): any two $\mathcal{K}$-extensions of a common model embed into a common one. Under amalgamation, Galois (orbital) types — equivalence classes of tuples under back-and-forth systems of partial $\mathcal{K}$-embeddings — behave like first-order types; $\mathrm{ga\text{-}S}(M)$ is the set of Galois types over $M$, and $\mathcal{K}$ is stable in $\mu$ if $|\mathrm{ga\text{-}S}(M)| \le \mu$ for all $M \in \mathcal{K}_\mu$.

The notions under comparison are local to a cardinal $\mu$. A type $p \in \mathrm{ga\text{-}S}(N)$ does not $\mu$-split over $M \le N$ if its restrictions to any two models of size $\mu$ between $M$ and $N$ agree under any isomorphism fixing $M$; $\mu$-splitting is the AEC analogue of a weak dividing notion used to measure dependence. $\mathcal{K}$ is $\mu$-superstable if $\mathcal{K}_\mu$ has joint embedding and no maximal models, $\mathcal{K}$ is stable in $\mu$, and $\mu$-splitting satisfies the 'no long splitting chains' property: for every increasing chain $\langle M_i : i < \alpha\rangle$, $\alpha < \mu^+$, of models of size $\mu$ with $M_{i+1}$ universal over $M_i$, every type over the union does not $\mu$-split over some $M_i$. $\mathcal{K}$ is $\mu$-tame if any two distinct Galois types over any model are already distinguished over some submodel of size at most $\mu$; tameness is a locality property for types that is automatic in elementary classes but not in general. A $(\mu,\delta)$-limit model over $M$ is the union of an increasing chain of length $\delta$ of models of size $\mu$, each universal (in the sense of embedding all $\mu$-sized extensions of its predecessor) over its predecessor; limit models are the AEC surrogate for saturated models when compactness fails. A superlimit model of size $\mu$ (Shelah) is a model that is universal in $\mathcal{K}_\mu$, has a proper $\mathcal{K}$-extension, and is isomorphic to the union of any increasing chain of copies of itself of length below $\mu^+$. A good $\mu$-frame is Shelah's localized forking calculus for types of singletons over models of size $\mu$: an independence relation satisfying invariance, monotonicity, extension over universal extensions, uniqueness, local character, and symmetry; it is type-full when its basic types over each model of $\mathcal{K}_\mu$ are all the non-algebraic Galois types over that model, and categorical when the underlying class $\mathcal{K}_\mu$ has a single model up to isomorphism. Good frames are the main engine of Shelah's categoricity transfers.

The symmetry property for non-$\mu$-splitting was isolated by Monica VanDieren: $\mathcal{K}$ has $\mu$-symmetry if whenever $M, M_0, N \in \mathcal{K}_\mu$ with $M$ universal over $M_0$ and $M_0$ a limit model over $N$, and elements $a, b$ are such that $a \in M \setminus M_0$, the type of $a$ over $M_0$ is non-algebraic and does not $\mu$-split over $N$, and the type of $b$ over $M$ is non-algebraic and does not $\mu$-split over $M_0$, then there is a limit model $M_b$ over $M_0$ containing $b$ such that the type of $a$ over $M_b$ does not $\mu$-split over $N$. Symmetry is the key input for uniqueness of limit models and for building frames from splitting. VanDieren–Vasey proved a downward transfer of symmetry, derived symmetry from failure of the order property, and showed that $\mu$-superstability with $\mu$-tameness implies $\mu$-symmetry, sharpening earlier chain-of-saturated-models results of Will Boney and Sebastien Vasey; the source also introduced four variants of $\mu$-symmetry — uniform, weak uniform, non-uniform, weak non-uniform — with uniform implying each of weak uniform and non-uniform, non-uniform implying weak non-uniform, uniform equivalent to the $\mu$-symmetry above, and weak uniform equivalent to uniform under $\mu$-superstability. The symmetry axiom of a type-full categorical good $\mu$-frame is equivalent to the weakest variant (weak non-uniform). Separately, Grossberg and VanDieren conjectured (2006) that categoricity in a high-enough cardinal implies tameness; what was known to follow unconditionally is only weak tameness, i.e. determination of types over saturated models, via Shelah's categoricity results.

Problem Statement

Let $\mathcal{K}$ be an abstract elementary class with amalgamation and let $\mu \ge \operatorname{LS}(\mathcal{K})$. Determine which of the following implications hold in general:

(1) (Question 4.13 of the source) If there exists a type-full categorical good $\mu$-frame on $\mathcal{K}_\mu$, must $\mathcal{K}$ have $\mu$-symmetry? More generally, if $\mathcal{K}$ is $\mu$-superstable — with no tameness or locality of Galois types assumed — must $\mathcal{K}$ have $\mu$-symmetry?

(2) (Symmetry variants) Among the four variants of $\mu$-symmetry (uniform, weak uniform, non-uniform, weak non-uniform), which of the proved implications can be reversed in general? Does every variant follow from $\mu$-superstability alone?

(3) (Question 6.12 of the source) If $\mathcal{K}$ is $\mu$-tame and there is a type-full good $\mu$-frame on $\mathcal{K}_\mu$ — or even only that $\mathcal{K}$ is $\mu$-superstable — is there necessarily a superlimit model of size $\mu$? Equivalently, assuming $\mu$-superstability and uniqueness of limit models of size $\mu$: is the union of an increasing chain of limit models in $\mathcal{K}_\mu$ of length less than $\mu^+$ itself a limit model?

(4) (Grossberg–VanDieren Conjecture 1.5) If $\mathcal{K}$ is categorical in some sufficiently large cardinal, must $\mathcal{K}$ be tame — that is, must Galois types over all models, and not merely over saturated models (where only weak tameness is known to follow), be determined by their restrictions to small submodels?

The common scientific objective is to establish the exact implication structure between the hypothesis side of the superstable theory of AECs with amalgamation ($\mu$-superstability, type-full categorical good $\mu$-frames, categoricity in a high-enough cardinal) and its structural-conclusion side (symmetry of non-$\mu$-splitting and its variants, existence of superlimit models of size $\mu$, tameness). Each of the four implications is to be decided in ZFC, with amalgamation only, without tameness, continuity of nonsplitting, or cardinal-arithmetic assumptions, and for every $\mu \ge \operatorname{LS}(\mathcal{K})$.

References

1. Monica M. VanDieren and Sebastien Vasey, Symmetry in abstract elementary classes with amalgamation, Archive for Mathematical Logic 56 (2017), no. 3-4, 423-452. DOI: 10.1007/s00153-017-0533-z. https://doi.org/10.1007/s00153-017-0533-z (arXiv:1508.03252)
2. Rami Grossberg and Monica VanDieren, Categoricity from one successor cardinal in tame abstract elementary classes, Journal of Mathematical Logic 6 (2006), no. 2, 181-201. DOI: 10.1142/S0219061306000554. https://doi.org/10.1142/S0219061306000554
3. Monica M. VanDieren, Superstability and symmetry, Annals of Pure and Applied Logic 167 (2016), no. 12, 1171-1183. DOI: 10.1016/j.apal.2016.05.004. https://doi.org/10.1016/j.apal.2016.05.004
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