[#P3098] Shelah's eventual categoricity conjecture for AECs

Problem. For every abstract elementary class \(K\) with Loewenheim-Skolem number \(\kappa\) and arbitrarily large models, is there a cardinal \(H=H(\kappa)\) such that categoricity of \(K\) in one \(\lambda\ge H\) implies categoricity in every \(\lambda'\ge H\)?

Models becoming categorical above a cardinal threshold. — A structural automaton diagram of the statement's mathematical objects.

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Definitions and notation

1Context

Known frontier: The conjecture holds for important tame or universal settings and under strong large-cardinal assumptions. Open boundary: The general ZFC theorem for arbitrary AECs remains open.

2Problem setup

Definition 1 (abstract elementary class). A class of structures with a coherent strong-substructure relation satisfying directed-union and Löwenheim-Skolem axioms.

Definition 2 (categorical in λ). Exactly one model of cardinality λ up to isomorphism.

Remark 1. Categoricity means uniqueness of the model at a given size. Morley's theorem gives an eventual transfer for first-order theories; the conjecture asks for its broad abstract-elementary analogue.

3What counts as a solution

Prove the stated tail transfer in ZFC with an explicit or definable threshold H(κ).
Or construct an AEC with arbitrarily large models categorical once above the proposed threshold but failing categoricity later.

1Status

Saved packet · August 1, 2026

What counts as a solution

Saved packet status (Current status and exact unresolved remainder). OPEN as checked on 2026-08-01. Strongest checked neighboring result: The conjecture holds for important tame or universal settings and under strong large-cardinal assumptions. Exact unresolved remainder: The general ZFC theorem for arbitrary AECs remains open.^[1][2]

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Record	Kind	Assessment
OPEN as checked on 2026-08-01. Strongest checked neighboring result: The conjecture holds for important tame or universal settings and under strong large-cardinal assumptions. Exact unresolved remainder: The general ZFC theorem for arbitrary AECs remains open.	Result	Evidence: Supported Backed by a cited source or by evidence short of a proof.
claim · Claim 1 OPEN as checked on 2026-08-01. Strongest checked neighboring result: The conjecture holds for important tame or universal settings and under strong large-cardinal assumptions. Exact unresolved remainder: The general ZFC theorem for arbitrary AECs remains open.^[1][2] Relevance to this problem This is the dated publication status for the canonical target Shelah's eventual categoricity conjecture for AECs. Evidence SupportedBacked by a cited source or by evidence short of a proof. Record state reported Scope No narrower scope recorded Sources Reference 1, Reference 2 Details The problem was checked as open on 2026-08-01. The strongest neighboring result found in the cited sources is: The conjecture holds for important tame or universal settings and under strong large-cardinal assumptions. The exact unresolved remainder is: The general ZFC theorem for arbitrary AECs remains open. A complete resolution must meet the following acceptance conditions: - Prove the stated tail transfer in ZFC with an explicit or definable threshold H(κ). - Or construct an AEC with arbitrarily large models categorical once above the proposed threshold but failing categoricity later. Open full record →
The conjecture holds for important tame or universal settings and under strong large-cardinal assumptions.	Result	Evidence: Supported Backed by a cited source or by evidence short of a proof.
claim · Claim 2 The conjecture holds for important tame or universal settings and under strong large-cardinal assumptions.^[1][2] Relevance to this problem Locates the present research frontier immediately below Shelah's eventual categoricity conjecture for AECs. Evidence SupportedBacked by a cited source or by evidence short of a proof. Record state reported Scope No narrower scope recorded Sources Reference 1, Reference 2 Details The conjecture holds for important tame or universal settings and under strong large-cardinal assumptions. This leaves the following boundary unresolved: The general ZFC theorem for arbitrary AECs remains open. The distinction is retained here so a restricted theorem, finite computation, or neighboring case is not presented as a solution of the full target. Related: Claim 1 Open full record →
The exact target, equivalent terminology, and 2025-2026 status evidence were checked on 2026-08-01. Strongest checked result: The conjecture holds for important tame or universal settings and under strong large-cardinal assumptions. Unresolved remainder: The general ZFC theorem for arbitrary AECs remains open.	Trace	Evidence: Supported Backed by a cited source or by evidence short of a proof.
attempt · Route 1 The exact target, equivalent terminology, and 2025-2026 status evidence were checked on 2026-08-01. Strongest checked result: The conjecture holds for important tame or universal settings and under strong large-cardinal assumptions. Unresolved remainder: The general ZFC theorem for arbitrary AECs remains open.^[1][2] Relevance to this problem Documents why Shelah's eventual categoricity conjecture for AECs was treated as a distinct open target on 2026-08-01. Evidence SupportedBacked by a cited source or by evidence short of a proof. Record state completed Scope No narrower scope recorded Sources Reference 1, Reference 2 What happened The audit ran on 2026-08-01 across the cited primary literature and problem collections, Crossref, arXiv, and stable publisher records, TheoremDB published, prospecting, packet, formal, and retired corpus. It checked the exact statement, its named or normalized variants, recent status evidence, and the controlled TheoremDB corpus. Queries included: - Shelah eventual categoricity conjecture AEC open 2026 - eventual categoricity abstract elementary classes large cardinals 2024 The exact target, equivalent terminology, and 2025-2026 status evidence were checked on 2026-08-01. Strongest checked result: The conjecture holds for important tame or universal settings and under strong large-cardinal assumptions. Unresolved remainder: The general ZFC theorem for arbitrary AECs remains open. Related: Claim 1 Open full record →
The general ZFC theorem for arbitrary AECs remains open.	Trace	Outcome: Open Evidence: Reported Stated by one agent or source, not independently checked.
attempt · Route 2 The general ZFC theorem for arbitrary AECs remains open. Relevance to this problem Turns the remaining uncertainty in Shelah's eventual categoricity conjecture for AECs into a checkable research target. Evidence ReportedStated by one agent or source, not independently checked. Record state open strategy Scope No narrower scope recorded What happened Research should address this boundary directly: The general ZFC theorem for arbitrary AECs remains open. A claimed resolution should satisfy every item below and preserve the statement's exact quantifiers and normalizations: - Prove the stated tail transfer in ZFC with an explicit or definable threshold H(κ). - Or construct an AEC with arbitrarily large models categorical once above the proposed threshold but failing categoricity later. Related: Claim 1 Open full record →
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Notes and companion material

Original intake status. OPEN as checked on 2026-08-01. Strongest checked neighboring result: The conjecture holds for important tame or universal settings and under strong large-cardinal assumptions. Exact unresolved remainder: The general ZFC theorem for arbitrary AECs remains open.

Equivalent-formulation queries: Shelah eventual categoricity conjecture AEC open 2026; eventual categoricity abstract elementary classes large cardinals 2024
Strongest checked neighboring result: The conjecture holds for important tame or universal settings and under strong large-cardinal assumptions.
Exact unresolved remainder: The general ZFC theorem for arbitrary AECs remains open.

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ProblemShelah's eventual categoricity conjecture for AECs

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Plain text

“Shelah's eventual categoricity conjecture for AECs.” TheoremDB. P3098. Problem statement; statement text SHA-256 f6f435841a2c0cd8b429540bb013147838859e7b03f7c0328c79a3bec842a777. https://theoremdb.org/statement/?ref=P3098

BibTeX

@misc{theoremdb-problem-f6f435841a2c0cd8b429540bb013147838859e7b03f7c0328c79a3bec842a777,
  title = {{Shelah's eventual categoricity conjecture for AECs}},
  howpublished = {TheoremDB},
  note = {Problem statement; statement text SHA-256 f6f435841a2c0cd8b429540bb013147838859e7b03f7c0328c79a3bec842a777},
  url = {https://theoremdb.org/statement/?ref=P3098}
}

Plain text: Built Markdown snapshot

This problem includes 4 records joined by 3 typed links, sourced from shelah.logic.at^[1], current as of August 1, 2026.

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1References

Packet source. S. Shelah and S. Vasey, Categoricity and multidimensional diagrams, Sh:842 (2024). eventual categoricity theorem under strongly compact cardinals. eventual categoricity theorem under strongly compact cardinals. ↗journal article · primary source · checked 2026-08-01Source use: original summary.Proves the conjectural behavior assuming a proper class of strongly compact cardinals.Also cited at S. Shelah and S. Vasey, Categoricity and multidimensional diagrams, Sh:842 (2024). eventual categoricity theorem under strongly compact cardinals.Source used to assess the problem's recorded status.For Shelah's eventual categoricity conjecture for AECs: This is the dated publication status for the canonical target Shelah's eventual categoricity conjecture for AECs.Source named by the research packet.
Sebastien Vasey, “Shelah's eventual categoricity conjecture in universal classes: part I”. Annals of Pure and Applied Logic 168 (2017), no. 9, 1609-1642. DOI 10.1016/j.apal.2017.03.003. arXiv:1506.07024 (2015). Conjecture 1.2 and main theorem. ↗preprint · primary source · arXiv:1506.07024, checked 2026-08-01 · checked 2026-08-01Source use: original summary.Proves substantial transfer results for universal classes and states the general AEC conjecture.Source used to assess the problem's recorded status.For Shelah's eventual categoricity conjecture for AECs: Proves substantial transfer results for universal classes and states the general AEC conjecture.

Original TheoremDB editorial statement and source synthesis; external works are used for citation only.

[#P3098] Shelah's eventual categoricity conjecture for AECs

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