For an abstract elementary class (AEC) with μ\mu-amalgamation and μ\mu-stability, Boney and VanDieren conjectured (Conjecture 1.1 of their Mathematical Logic Quarterly 70 (2024) paper, circulating since 2015) that the set of regular ordinals α<μ+\alpha<\mu^{+} for which every (μ,α)(\mu,\alpha)-limit model is isomorphic to the (μ,μ)(\mu,\mu)-limit model forms a non-trivial interval of regular cardinals whose minimum is κμ∗(K)\kappa^{*}_{\mu}(\mathcal{K}), the universal local character for μ\mu-nonsplitting — the AEC analogue of the first-order superstability threshold κ(T)\kappa(T). Their Theorem 1.2 proves the upper end of the conjecture under additional continuity and symmetry hypotheses for non-μ\mu-splitting. Later work resolved major special cases — the first-order case completely (Beard 2025) and the ℵ0\aleph_{0}-tame case with a forking-like independence relation (Beard–Mazari-Armida 2025), where the spectrum is exactly [κλ∗(K),λ+)∩Reg[\kappa^{*}_{\lambda}(\mathbf{K}),\lambda^{+})\cap\mathrm{Reg} — and Beard (2025) weakened the hypotheses needed for the isomorphism half to a splitting-like relation satisfying only a weak symmetry input. The conjecture at its stated generality — no tameness, no pre-existing independence notion, no symmetry or continuity assumptions — remains open: deriving the symmetry/continuity inputs from μ\mu-stability and amalgamation alone (Leung's 2024 attempt at this contains a documented gap), removing tameness from the non-isomorphism half (explicitly posed as Problem 4.15 by Beard–Mazari-Armida), and bounding κμ∗(K)<μ+\kappa^{*}_{\mu}(\mathcal{K})<\mu^{+} in ZFC without tameness are all unresolved.

Background

An abstract elementary class (AEC) is a pair K=(K,≤K)\mathcal{K}=(K,\leq_{\mathcal{K}}) consisting of a class of structures in a fixed language, closed under isomorphisms, together with a 'strong submodel' relation ≤K\leq_{\mathcal{K}} strengthening ordinary substructure, satisfying abstracted Löwenheim–Skolem and chain-union axioms. Elementary classes of complete first-order theories with elementary embedding are the prototype, but AECs also capture classes defined in infinitary logics or by algebraic closure conditions (for example, modules with pure embeddings). The Löwenheim–Skolem number LS(K)\mathrm{LS}(\mathcal{K}) plays the role that ∣T∣|T| plays for a theory. Types in an AEC are Galois (orbital) types: ga-tp(a/M)\mathrm{ga\text{-}tp}(a/M) identifies two realizations if an automorphism of a sufficiently large ambient model carries one to the other fixing MM. K\mathcal{K} is μ\mu-stable if every model of size μ\mu carries at most μ\mu Galois types, and has μ\mu-amalgamation if any two extensions of a model of size μ\mu inside Kμ\mathcal{K}_{\mu} amalgamate over it.

Limit models are the AEC surrogate for saturated models. A model NN is universal over MM (of the same size) if every extension of MM of that size embeds into NN over MM; a (μ,α)(\mu,\alpha)-limit model, for a limit ordinal α<μ+\alpha<\mu^{+}, is the union of an increasing continuous chain ⟨Mi:i<α⟩\langle M_{i}:i<\alpha\rangle of models of size μ\mu with each Mi+1M_{i+1} universal over MiM_{i}. Under μ\mu-stability (with amalgamation and no maximal models) limit models exist, and two limit models of the same cofinality are always isomorphic by a back-and-forth argument; the substantive question is when limit models of different cofinalities are isomorphic. Because limit models measure how much saturation can be built internally at a single cardinal, their isomorphism types are a basic structure ingredient of the classification theory for AECs that feeds Shelah's categoricity conjecture, and uniqueness of limit models underlies the construction of forking-like independence relations beyond the first-order setting.

In a complete first-order stable theory TT, κ(T)\kappa(T) is the least cardinal κ\kappa such that for every ⊆\subseteq-increasing chain of sets ⟨Ai:i≤κ⟩\langle A_{i}:i\leq\kappa\rangle and every type p∈S(⋃i<κAi)p\in S(\bigcup_{i<\kappa}A_{i}), pp does not fork over some AiA_{i} (local character of forking), and κr(T)\kappa_{r}(T) is the first regular cardinal ≥κ(T)\geq\kappa(T). If TT is superstable (κ(T)=ℵ0\kappa(T)=\aleph_{0}) then all limit models of a given size are isomorphic; in a strictly stable theory (κ(T)>ℵ0\kappa(T)>\aleph_{0}) the (λ,ℵ0)(\lambda,\aleph_{0})-limit model is not isomorphic to the (λ,μ)(\lambda,\mu)-limit model for any regular μ≥κr(T)\mu\geq\kappa_{r}(T) (Grossberg–VanDieren–Villaveces 2016, Example 6.1), and Beard (2025) completed the first-order picture: for λ≥∣T∣+ℵ0\lambda\geq|T|+\aleph_{0} with TT λ\lambda-stable, the (λ,δ1)(\lambda,\delta_{1})- and (λ,δ2)(\lambda,\delta_{2})-limit models (cf⁡(δ1)<cf⁡(δ2)\operatorname{cf}(\delta_{1})<\operatorname{cf}(\delta_{2})) are isomorphic if and only if cf⁡(δ1)≥κr(T)\operatorname{cf}(\delta_{1})\geq\kappa_{r}(T), giving exactly ∣α∣+1|\alpha|+1 isomorphism types when κr(T)=ℵα\kappa_{r}(T)=\aleph_{\alpha}.

For AECs, the superstable analogue of this uniqueness was established through a chain of works (VanDieren 2016a,b; Grossberg–VanDieren–Villaveces 2016; VanDieren–Vasey 2017), where λ\lambda-superstability and λ\lambda-symmetry were the key hypotheses; VanDieren–Vasey (2017, Corollary 6.9) later derived μ\mu-symmetry from μ\mu-superstability together with μ\mu-tameness (amalgamation assumed throughout), and their Question 4.13 explicitly leaves open whether μ\mu-superstability alone suffices. Strictly stable AECs — stable in μ\mu but not μ\mu-superstable — are the frontier, especially without tameness (tameness is the locality property that Galois types over a model are determined by their restrictions to small submodels; most AEC classification theory since Grossberg–VanDieren has relied on it). Boney–VanDieren (2024) introduced κμ∗(K)\kappa^{*}_{\mu}(\mathcal{K}), the universal local character for μ\mu-nonsplitting: the minimal regular κ<μ+\kappa<\mu^{+} such that along every increasing continuous chain ⟨Mi∈Kμ:i≤α⟩\langle M_{i}\in\mathcal{K}_{\mu}:i\leq\alpha\rangle of regular length α≥κ\alpha\geq\kappa with Mi+1M_{i+1} universal over MiM_{i}, every non-algebraic Galois type over MαM_{\alpha} — non-algebraic meaning that the realization is not already an element of the model over which the type is taken — does not μ\mu-split over some MiM_{i} (κμ∗(K)=∞\kappa^{*}_{\mu}(\mathcal{K})=\infty if no such κ\kappa exists). Here a type pp over MM μ\mu-splits over N≤KMN\leq_{\mathcal{K}}M if two submodels of MM extending NN that are isomorphic over NN disagree about pp; non-splitting is essentially the only independence-like notion available without tameness. κμ∗(K)=ℵ0\kappa^{*}_{\mu}(\mathcal{K})=\aleph_{0} recovers μ\mu-superstability, and larger values quantify 'how far from superstable' the class is, generalizing κ(T)\kappa(T). Under (λ,μ)(\lambda,\mu)-tameness, λ\lambda-stability and amalgamation one has κμ∗(K)≤λ\kappa^{*}_{\mu}(\mathcal{K})\leq\lambda (Boney–VanDieren, Proposition 2.2), but no ZFC bound is known without tameness or categoricity-type hypotheses.

Boney–VanDieren's Theorem 1.2 proves the upper end of their conjecture under Assumption 2.3 — joint embedding, amalgamation and no maximal models in Kμ\mathcal{K}_{\mu}; μ\mu-stability; κμ∗(K)<μ+\kappa^{*}_{\mu}(\mathcal{K})<\mu^{+}; and limit continuity for non-μ\mu-splitting (if pp is a type over a limit model witnessed by ⟨Mi:i<θ⟩\langle M_{i}:i<\theta\rangle and each p ⁣↾ ⁣Mip\!\upharpoonright\!M_{i} does not μ\mu-split over NN, then pp does not μ\mu-split over NN) — plus symmetry for non-μ\mu-splitting (non-splitting independence is symmetric in its two arguments; they use a localized version called (μ,δ)(\mu,\delta)-symmetry). All proofs in this program go through towers: increasing chains of models with distinguished elements recording independence, a technique introduced by Shelah–Villaveces and developed by VanDieren, Vasey, and others.

In 2025 the picture moved substantially. Beard–Mazari-Armida (Annals of Pure and Applied Logic 176 (2025)) determined the full spectrum under ℵ0\aleph_{0}-tameness plus the existence of a forking-like independence relation ⌢\smallfrown on Kλ\mathbf{K}_{\lambda} satisfying uniqueness, extension, non-forking amalgamation, universal continuity, and (≥κ)(\geq\kappa)-local character: the set Γ\Gamma of regular α<λ+\alpha<\lambda^{+} such that the (λ,α)(\lambda,\alpha)-limit model is isomorphic to the (λ,λ)(\lambda,\lambda)-limit model is exactly [κ(⌢,Kλ,≤u),λ+)∩Reg[\kappa(\smallfrown,\mathbf{K}_{\lambda},\leq^{u}),\lambda^{+})\cap\mathrm{Reg}, where ≤u\leq^{u} is the 'universal over' ordering and κ(⌢,Kλ,≤u)\kappa(\smallfrown,\mathbf{K}_{\lambda},\leq^{u}) denotes the local-character threshold of ⌢\smallfrown along ≤u\leq^{u}-chains, and by their Corollary 4.8 the minimum equals the universal local character of non-splitting — the κ∗\kappa^{*}-threshold — so Conjecture 1.1 follows under these hypotheses (their Corollary 5.4). Notably their isomorphism ('long limit') half assumes no tameness and no symmetry axiom for the relation itself — the symmetry-type input enters through its non-forking amalgamation hypothesis, which Beard later weakened — while their non-isomorphism ('short limit') half uses ℵ0\aleph_{0}-tameness; they explicitly ask (Problem 4.15) whether tameness can be removed. Beard (arXiv:2511.18665) then weakened the requirements for the isomorphism half to a splitting-like relation with weak uniqueness, weak existence, universal continuity, (≥κ)(\geq\kappa)-local character, and a (λ,θ)(\lambda,\theta)-weak non-forking amalgamation (a weakened symmetry), reobtaining Boney–VanDieren's Theorem 1.2 and the Beard–Mazari-Armida theorem as corollaries. On the other hand, Leung (Journal of Symbolic Logic 89 (2024)) had claimed to derive uniqueness of non-forking and symmetry — thereby removing the symmetry assumption from Boney–VanDieren's theorem — from tameness, a monster model, continuity of non-splitting, and stability on an interval of cardinals; Beard–Mazari-Armida (Remark 2.34) identified an error in Leung's uniqueness proof and provided a corrected proof only of the related canonicity statement under additional hypotheses, so that derivation remains open. The conjecture in its full generality — hypotheses only μ\mu-amalgamation and μ\mu-stability — is the open problem recorded here.

Problem Statement

Let K\mathcal{K} be an abstract elementary class and μ\mu a cardinal such that K\mathcal{K} has μ\mu-amalgamation (amalgamation for spans of models of size μ\mu) and is μ\mu-stable; under these hypotheses limit models of size μ\mu exist, so the following set is meaningful. Define

Γμ(K)={α<μ+  :  cf⁡(α)=α and every (μ,α)-limit model is isomorphic to the (μ,μ)-limit model}.\Gamma_{\mu}(\mathcal{K})=\left\{\alpha<\mu^{+}\;:\;\operatorname{cf}(\alpha)=\alpha\text{ and every } (\mu,\alpha)\text{-limit model is isomorphic to the }(\mu,\mu)\text{-limit model}\right\}.

Boney–VanDieren's Conjecture 1.1 asserts that Γμ(K)\Gamma_{\mu}(\mathcal{K}) is a non-trivial interval of regular cardinals — an end segment of the form [κ,μ+)∩Reg[\kappa,\mu^{+})\cap\mathrm{Reg} with κ<μ+\kappa<\mu^{+} — and that its minimum is κμ∗(K)\kappa^{*}_{\mu}(\mathcal{K}), the universal local character for μ\mu-nonsplitting (the least regular κ<μ+\kappa<\mu^{+} such that along every increasing continuous chain of models of size μ\mu, of regular length ≥κ\geq\kappa, in which each successor is universal over its predecessor, every non-algebraic Galois type over the top model fails to μ\mu-split over some earlier stage; κμ∗(K)=∞\kappa^{*}_{\mu}(\mathcal{K})=\infty if no such κ\kappa exists, so the conjecture in particular entails κμ∗(K)<μ+\kappa^{*}_{\mu}(\mathcal{K})<\mu^{+}).

The problem is to prove or refute this conjecture in its stated generality, where only μ\mu-amalgamation and μ\mu-stability are hypothesized: prove that for every such K\mathcal{K} the spectrum Γμ(K)\Gamma_{\mu}(\mathcal{K}) is exactly [κμ∗(K),μ+)∩Reg[\kappa^{*}_{\mu}(\mathcal{K}),\mu^{+})\cap\mathrm{Reg}; or refute it by exhibiting an AEC with μ\mu-amalgamation and μ\mu-stability for which the conjectured structure fails — for example, limit models of two regular lengths κμ∗(K)≤α<β<μ+\kappa^{*}_{\mu}(\mathcal{K})\leq\alpha<\beta<\mu^{+} that are not isomorphic, a gap in the spectrum (regular α1<α2<α3\alpha_{1}<\alpha_{2}<\alpha_{3} with α1,α3∈Γ\alpha_{1},\alpha_{3}\in\Gamma but α2∉Γ\alpha_{2}\notin\Gamma), a minimum different from κμ∗(K)\kappa^{*}_{\mu}(\mathcal{K}), or a class with κμ∗(K)=∞\kappa^{*}_{\mu}(\mathcal{K})=\infty whose limit models are nonetheless unique.

The conjecture is already proved under ℵ0\aleph_{0}-tameness together with the existence of a forking-like independence relation (Beard–Mazari-Armida 2025, Corollary 5.4), and in the first-order case (Beard 2025). All current proofs consume a symmetry-type input and a continuity input for non-splitting; a full resolution bears on several documented open problems: whether these inputs follow from μ\mu-stability and amalgamation alone (Leung's 2024 attempt at deriving them contains a documented gap), whether the non-isomorphism half survives without tameness (Beard–Mazari-Armida's Problem 4.15), and whether κμ∗(K)<μ+\kappa^{*}_{\mu}(\mathcal{K})<\mu^{+} holds in ZFC for every μ\mu-stable AEC with μ\mu-amalgamation. Some known results assume extra set-theoretic hypotheses, such as Jensen's diamond principle (a combinatorial guessing principle implying the continuum hypothesis) or the singular cardinal hypothesis (a restriction on cardinal exponentiation at singular cardinals).

References

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