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arxiv: 2405.08211 · v3 · submitted 2024-05-13 · 🧮 math.LO

Simple Homogeneous Structures and Indiscernible Sequence Invariants

John Baldwin , James Freitag , Scott Mutchnik This is my paper

Pith reviewed 2026-05-24 00:47 UTC · model grok-4.3

classification 🧮 math.LO
keywords simple theoriesone-basedfinite rankindiscernible sequencesKim-forkingNSOP1quantifier eliminationhomogeneous structures
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The pith

Every simple theory with quantifier elimination in a finite relational language has finite rank and is one-based.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper introduces invariants F_ind and its dual F_Mb for dependence in indiscernible sequences, along with the definable Morley property and n-resolvability. These are applied to show that the degree of nonminimality can take any positive integer value in an ω-stable theory. The central result proves a conjecture that every simple theory with quantifier elimination in a finite relational language has finite rank and is one-based, relying on types with infinite F_Mb and n-resolvability. Variants of the simple Kim-forking conjecture are established in NSOP1 theories, and F_Mb is shown to be nontrivial in stable theories.

Core claim

We introduce some properties describing dependence in indiscernible sequences: F_ind and its dual F_Mb, the definable Morley property, and n-resolvability. We show that the degree of nonminimality may take on any positive integer value in an ω-stable theory. Proving a conjecture of Koponen, we show that every simple theory with quantifier elimination in a finite relational language has finite rank and is one-based. The arguments closely rely on finding types q with F_Mb(q) = ∞, and on n-resolvability. We prove some variants of the simple Kim-forking conjecture in NSOP1 theories. We show that the quantity F_Mb is in fact nontrivial even in stable theories.

What carries the argument

The invariants F_ind and F_Mb on indiscernible sequences, which quantify dependence along such sequences in simple and NSOP1 theories.

If this is right

  • The degree of nonminimality can take any positive integer value in ω-stable theories.
  • Every simple theory with quantifier elimination in a finite relational language has finite rank and is one-based.
  • A global analogue of the simple Kim-forking conjecture holds with infinitely many variables in every NSOP1 theory.
  • Kim-forking satisfies finite-variable versions when F_Mb(p) is finite.
  • F_Mb takes nontrivial values in stable theories.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The new invariants could be used to study dependence in other classes of theories not covered in the paper.
  • The classification might imply that all homogeneous structures in such simple theories are of bounded complexity.
  • Further study of n-resolvability could reveal when these classification results apply more broadly.

Load-bearing premise

The classification proofs require the existence of types q with F_Mb(q) = infinity and the property of n-resolvability.

What would settle it

A simple theory with quantifier elimination in a finite relational language that has infinite rank or is not one-based would falsify the main claim.

read the original abstract

We introduce some properties describing dependence in indiscernible sequences: $F_{ind}$ and its dual $F_{Mb}$, the definable Morley property, and $n$-resolvability. Applying these properties, we establish the following results: We show that the degree of nonminimality introduced by Freitag and Moosa, which is closely related to $F_{ind}$ (equal in $\mathrm{DCF}_{0}$), may take on any positive integer value in an $\omega$-stable theory, answering a question of Freitag, Jaoui, and Moosa. Proving a conjecture of Koponen, we show that every simple theory with quantifier elimination in a finite relational language has finite rank and is one-based. The arguments closely rely on finding types $q$ with $F_{Mb}(q) = \infty$, and on $n$-resolvability. We prove some variants of the simple Kim-forking conjecture, a generalization of the stable forking conjecture to $\mathrm{NSOP}_{1}$ theories. We show a global analogue of the simple Kim-forking conjecture with infinitely many variables holds in every $\mathrm{NSOP}_{1}$ theory, and show that Kim-forking with a realization of a type $p$ with $\mathrm{F}_{Mb}(p) < \infty$ satisfies a finite-variable version of this result. We then show, in a low $\mathrm{NSOP}_{1}$ theory or when $p$ is isolated, if $p \in S(C)$ has the definable Morley property for Kim-independence, Kim-forking with realizations of $p$ gives a nontrivial instance of the simple Kim-forking conjecture itself. In particular, when $F_{Mb}(p) < \infty$ and $|S^{F_{Mb}(p) + 1}(C)| < \infty$, Kim-forking with realizations of $p$ gives us a nontrivial instance of the simple Kim-forking conjecture. We show that the quantity $F_{Mb}$, motivated in simple and $\mathrm{NSOP}_{1}$ theories by the above results, is in fact nontrivial even in stable theories.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper introduces invariants F_ind and its dual F_Mb, the definable Morley property, and n-resolvability to describe dependence in indiscernible sequences. It applies these to show that the degree of nonminimality can take any positive integer value in an ω-stable theory; proves Koponen's conjecture that every simple theory with quantifier elimination in a finite relational language has finite rank and is one-based (relying on types q with F_Mb(q)=∞ and n-resolvability); establishes variants of the simple Kim-forking conjecture in NSOP1 theories, including a global infinite-variable version and finite-variable cases under definable Morley property or isolation; and shows F_Mb is nontrivial even in stable theories.

Significance. If the central arguments hold, the work supplies new invariants for indiscernible sequences that resolve an open conjecture in the classification of simple theories and yield partial progress on the simple Kim-forking conjecture. The explicit construction of examples realizing every positive integer degree of nonminimality and the demonstration that F_Mb remains nontrivial in stable theories are concrete contributions that strengthen the toolkit for studying forking and independence in model theory.

major comments (2)
  1. [Koponen conjecture proof] The section proving Koponen's conjecture: the argument that every simple theory with QE in a finite relational language has finite rank and is one-based rests on the existence of types q with F_Mb(q)=∞ together with n-resolvability; the manuscript must supply an explicit theorem or construction guaranteeing these two properties hold universally in the target class, as the abstract itself identifies this dependence and it is load-bearing for the classification claim.
  2. [Kim-forking variants] The subsection on finite-variable Kim-forking: the claim that Kim-forking with a realization of p where F_Mb(p)<∞ satisfies a finite-variable version of the simple Kim-forking conjecture requires a verification that the bound |S^{F_Mb(p)+1}(C)|<∞ does not collapse the instance to a trivial or previously known case; without this check the reduction from the global infinite-variable result is not fully substantiated.
minor comments (2)
  1. [Introduction] Notation for F_Mb and F_ind should be introduced with a short table comparing their definitions and duality properties to aid readability.
  2. [Stable theories section] The statement that F_Mb is nontrivial in stable theories would benefit from an explicit example (e.g., a specific stable theory and type) rather than a general existence argument.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive feedback on our manuscript. We address the two major comments point by point below, agreeing where revision is needed to strengthen the presentation.

read point-by-point responses

  1. Referee: [Koponen conjecture proof] The section proving Koponen's conjecture: the argument that every simple theory with QE in a finite relational language has finite rank and is one-based rests on the existence of types q with F_Mb(q)=∞ together with n-resolvability; the manuscript must supply an explicit theorem or construction guaranteeing these two properties hold universally in the target class, as the abstract itself identifies this dependence and it is load-bearing for the classification claim.

    Authors: We agree that the proof of Koponen's conjecture is load-bearing on the existence of types q with F_Mb(q)=∞ and on n-resolvability, as noted in the abstract. The manuscript constructs such types and verifies n-resolvability within the argument for the target class of theories, but we acknowledge that an explicit standalone theorem guaranteeing these properties universally would improve clarity and self-containment. We will add such a theorem (with its proof) in the revised version. revision: yes

  2. Referee: [Kim-forking variants] The subsection on finite-variable Kim-forking: the claim that Kim-forking with a realization of p where F_Mb(p)<∞ satisfies a finite-variable version of the simple Kim-forking conjecture requires a verification that the bound |S^{F_Mb(p)+1}(C)|<∞ does not collapse the instance to a trivial or previously known case; without this check the reduction from the global infinite-variable result is not fully substantiated.

    Authors: We agree that the finite-variable claim under the bound |S^{F_Mb(p)+1}(C)|<∞ requires explicit verification that the resulting instance remains nontrivial. The manuscript already notes that this yields a nontrivial instance when combined with the definable Morley property or isolation, but we will add a dedicated paragraph or example in the revised version confirming that the bound does not reduce to previously known trivial cases, thereby substantiating the reduction from the global result. revision: yes

Circularity Check

0 steps flagged

No circularity: new invariants introduced and applied to independent classification results

full rationale

The paper defines new dependence properties (F_ind, F_Mb, definable Morley property, n-resolvability) in the abstract and applies them to prove external conjectures and answer open questions. The Koponen conjecture proof is described as relying on locating types with F_Mb(q)=∞ and verifying n-resolvability, but these are presented as proof techniques rather than definitions that presuppose the target conclusion. No quoted step equates a derived quantity to a fitted input or reduces the central claim to a self-citation chain. The results on degree of nonminimality, Kim-forking variants, and F_Mb in stable theories are independent of the main classification and do not exhibit self-definitional or renaming patterns. The derivation remains self-contained against the stated assumptions.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 4 invented entities

The central claims rest on the new invariants F_ind, F_Mb, the definable Morley property, and n-resolvability being well-defined and useful in the relevant classes of theories; these are introduced in the paper rather than derived from prior literature.

axioms (1)
  • standard math Standard axioms of first-order model theory and the definition of simple and NSOP1 theories
    Invoked throughout to set the ambient context for the new invariants.
invented entities (4)
  • F_ind no independent evidence
    purpose: Measure of dependence along indiscernible sequences
    Newly introduced property; no independent evidence supplied in abstract.
  • F_Mb no independent evidence
    purpose: Dual measure to F_ind
    Newly introduced property; no independent evidence supplied in abstract.
  • definable Morley property no independent evidence
    purpose: Property of types with respect to Kim-independence
    Newly introduced property; no independent evidence supplied in abstract.
  • n-resolvability no independent evidence
    purpose: Technical condition used in proofs
    Newly introduced property; no independent evidence supplied in abstract.

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Higher-arity distality and forking triviality

    math.LO 2026-05 unverdicted novelty 6.0

    k-triviality collapses to 1-triviality among simple theories, yielding new non-k-ary examples of strongly k-distal theories and implying that certain stable theories are trivial.

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