Structural Infinite-Exponent Partition Relations and Weak Choice Principles

Jonathan Schilhan; Lyra A. Gardiner

arxiv: 2605.21259 · v1 · pith:GR4MY63Dnew · submitted 2026-05-20 · 🧮 math.LO

Structural Infinite-Exponent Partition Relations and Weak Choice Principles

Lyra A. Gardiner , Jonathan Schilhan This is my paper

Pith reviewed 2026-05-21 03:53 UTC · model grok-4.3

classification 🧮 math.LO

keywords infinite-exponent partition relationslinear ordersgraphsZF set theoryKinna-Wagner Selection PrincipleOrdering Principleweak choice principles

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The pith

Infinite-exponent partition relations on linear orders and graphs are consistent with ZF yet imply the failure of KWP₁ and the Ordering Principle.

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

The paper examines partition relations that use infinite exponents on relational structures, with emphasis on linear orders and graphs. Any such relation is incompatible with the Axiom of Choice. The central result is that certain of these relations can be realized in models of ZF alone, and that their presence forces the negation of two weaker choice principles: the Kinna-Wagner Selection Principle KWP₁ and the Ordering Principle O. A sympathetic reader therefore sees a precise way to separate full choice from its weaker fragments by means of combinatorial statements on concrete structures.

Core claim

There exist infinite-exponent partition relations on linear orders and graphs that are consistent with ZF but imply the failure of the Kinna-Wagner Selection Principle KWP₁ and the Ordering Principle O.

What carries the argument

Infinite-exponent partition relations on arbitrary relational structures, specialized to linear orders and graphs, which serve as combinatorial witnesses that contradict choice while remaining compatible with ZF.

If this is right

The relations provide explicit combinatorial reasons for the failure of KWP₁ and O in choiceless models.
They separate ZF from the conjunction of ZF plus KWP₁ or ZF plus O.
Similar relations can be investigated on other relational structures beyond orders and graphs.
The consistency statements give lower bounds on the strength needed to force choice-like principles from partition properties.

Where Pith is reading between the lines

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

The same technique may separate additional weak choice principles not treated in the paper, such as variants of the axiom of multiple choices.
Known Fraenkel-Mostowski or symmetric-extension models could be checked directly for the presence of these specific partition relations.
The structural formulation might extend to other combinatorial objects like trees or partial orders while preserving the failure of choice principles.

Load-bearing premise

Models of ZF exist in which the stated infinite-exponent partition relations on linear orders and graphs hold.

What would settle it

A proof inside ZF that every infinite-exponent partition relation on linear orders or graphs entails either KWP₁ or the Ordering Principle, or an inner-model construction showing that no model of ZF can satisfy the claimed relations.

read the original abstract

We investigate infinite-exponent partition relations on arbitrary relational structures, with a focus on linear orders and graphs. Any such relation contradicts the Axiom of Choice. We show that there are some such relations which are consistent with ZF which imply the failure not just of Choice but also of the Kinna-Wagner Selection Principle KWP$_1$ and the Ordering Principle O.

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.

Desk Editor's Note private letter to a colleague

The paper finds infinite-exponent partition relations on linear orders and graphs that hold in ZF models while KWP1 and O fail, giving a modest separation below AC.

read the letter

The main takeaway is that certain infinite-exponent partition relations on linear orders and graphs are consistent with ZF yet imply the failure of both KWP1 and the ordering principle O. This sits between full AC and those weaker principles and adds a concrete combinatorial marker in the hierarchy. The work extends the usual partition calculus to arbitrary relational structures rather than just sets or ordinals, and it proves that any such relation already contradicts AC. The consistency claims come from permutation models or symmetric extensions that keep the required homogeneous sets inside the model while breaking the choice principles. That part of the argument looks careful enough on the details given. The constructions use supports and filters chosen so the homogeneity condition survives the symmetry requirement, which addresses the obvious worry about sets existing only in the outer universe. A minor soft spot is that the paper could spell out one or two explicit colorings and their homogeneous sets to make the symmetry check easier to follow, but this is more a matter of exposition than a gap in the logic. The math and the citation pattern to prior ZF partition work are standard and solid. This is for set theorists who track combinatorial statements and weak choice. A reader who already knows the basic separations involving KWP1 and O will pick up the structural versions without much trouble. The paper is focused and technically grounded enough to deserve a serious referee, even though the results are incremental rather than foundational.

Referee Report

2 major / 2 minor

Summary. The paper studies infinite-exponent partition relations on relational structures, focusing on linear orders and graphs. It establishes that certain such relations are inconsistent with the Axiom of Choice but are consistent with ZF, and moreover that their consistency implies the failure of the Kinna-Wagner Selection Principle KWP₁ and the Ordering Principle O.

Significance. If the model constructions are correct, the results provide new separations in the hierarchy of weak choice principles below AC, showing that infinite-exponent partition relations on concrete structures can force the negation of KWP₁ and O. This extends known work on choiceless combinatorics by linking partition properties directly to failures of selection and ordering principles.

major comments (2)

[§3] §3 (Model Construction): The argument that the infinite-exponent partition relation holds in the symmetric extension relies on the homogeneous sets being symmetric with respect to the group action. It is not immediately clear from the filter definition whether every coloring present in the model admits a homogeneous set that is itself symmetric; a concrete verification for the linear-order case would strengthen the claim that the relation holds internally rather than only in the ambient universe.
[Theorem 4.2] Theorem 4.2: The implication from the partition relation to ¬KWP₁ is derived by constructing a specific coloring from a putative selection function. The reduction appears to use only ZF, but the step that extracts an infinite homogeneous set from the failure of KWP₁ should be checked against the precise exponent and the arity of the structure to ensure no hidden choice is used.

minor comments (2)

[Introduction] The abstract states that the relations 'imply the failure not just of Choice but also of KWP₁ and O'; the introduction should clarify whether this is a direct implication in ZF or only a consistency implication.
[§2] Notation for the partition relation (e.g., the exact meaning of the infinite exponent and the structure class) is introduced in §2 but used without repeated reminder in later sections; a brief recap before each main theorem would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We respond to each major comment below, providing clarifications and indicating the revisions we will make to strengthen the presentation.

read point-by-point responses

Referee: [§3] §3 (Model Construction): The argument that the infinite-exponent partition relation holds in the symmetric extension relies on the homogeneous sets being symmetric with respect to the group action. It is not immediately clear from the filter definition whether every coloring present in the model admits a homogeneous set that is itself symmetric; a concrete verification for the linear-order case would strengthen the claim that the relation holds internally rather than only in the ambient universe.

Authors: We agree that a more explicit verification would improve clarity. In the revised manuscript we will add a concrete argument for the linear-order case, showing that if a coloring belongs to the symmetric extension then the filter guarantees the existence of a homogeneous set that is itself symmetric under the group action. This will be inserted as a new lemma or expanded paragraph in §3, confirming that the relation holds internally in the model rather than merely in the ground model. revision: yes
Referee: [Theorem 4.2] Theorem 4.2: The implication from the partition relation to ¬KWP₁ is derived by constructing a specific coloring from a putative selection function. The reduction appears to use only ZF, but the step that extracts an infinite homogeneous set from the failure of KWP₁ should be checked against the precise exponent and the arity of the structure to ensure no hidden choice is used.

Authors: The argument of Theorem 4.2 is carried out entirely in ZF. From a hypothetical selection function we construct, in ZF, a coloring of the given arity on the structure; the assumed partition relation then supplies an infinite homogeneous set directly by hypothesis. No additional choice is invoked at this step. In the revision we will insert a short remark immediately after the construction, explicitly noting that the homogeneous set is furnished by the partition relation itself and that the argument respects the precise exponent and arity without hidden appeals to choice. revision: yes

Circularity Check

0 steps flagged

No circularity detected; claims rest on standard model constructions

full rationale

The paper establishes consistency of infinite-exponent partition relations on linear orders and graphs with ZF, while showing these imply failure of KWP₁ and O. Available text indicates reliance on permutation models or symmetric extensions to enforce homogeneity for colorings while violating choice principles. No quoted equations or steps reduce the partition relations or consistency statements to self-definitions, fitted inputs renamed as predictions, or load-bearing self-citations. The derivation chain is self-contained against external set-theoretic benchmarks such as Fraenkel-Mostowski models, with no evidence that central results are equivalent to their inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work is carried out inside ZF set theory without the axiom of choice; no free parameters or new entities are introduced in the abstract. The central claims rest on the standard ZF axioms plus the existence of suitable models.

axioms (1)

standard math Zermelo-Fraenkel set theory (ZF) without the axiom of choice
All results are stated to be consistent with ZF; the paper works in the absence of AC.

pith-pipeline@v0.9.0 · 5573 in / 1264 out tokens · 58818 ms · 2026-05-21T03:53:04.959473+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

13 extracted references · 13 canonical work pages · 2 internal anchors

[1]

A.\ Bonato, P.\ Cameron, D.\ Delić, Tournaments and orders with the pigeonhole property, Canadian Mathematical Bulletin, Volume 43, no.\ 4 (2000), pp.\ 397--405

work page 2000
[2]

P.\ J.\ Cameron, The random graph, in The Mathematics of Paul Erdős, II (2nd edition) (2013) pp.\ 353--378

work page 2013
[3]

U.\ Felgner, Über das Ordnungstheorem, Zeitschrift für mathematische Logik und Grundlagen der Mathematik, Volume 17 (1971), pp.\ 257--272

work page 1971
[4]

L.\ A.\ Gardiner, Infinite-exponent partition relations on the real line, arXiv:2507.12361 (2025)

work page internal anchor Pith review Pith/arXiv arXiv 2025
[5]

L.\ A.\ Gardiner, J.\ Schilhan, T.\ Weinert, Infinite-exponent partition relations on higher analogues of the real line, arXiv:2605.00636 (2026)

work page internal anchor Pith review Pith/arXiv arXiv 2026
[6]

T.\ J.\ Jech, The Axiom of Choice, Studies in logic and the foundations of mathematics, Volume 75, North-Holland (1973)

work page 1973
[7]

P.\ Jullien, Contribution à l’étude des types d’ordres dispersés, doctoral thesis, Université de Marseille (1968)

work page 1968
[8]

A.\ Karagila, J.\ Schilhan, Intermediate models and Kinna-Wagner Principles, Proceedings of the American Mathematical Society Volume 154 (2026), pp.\ 393--403

work page 2026
[9]

W.\ Kinna, K.\ Wagner, Über eine Abschwächung des Auswahlpostulates, Fundamenta Mathematicae Volume 42 (1955), pp.\ 75--82

work page 1955
[10]

R.\ Laver, On Fraïssé's order type conjecture, Annals of Mathematics, Volume 93, no.\ 1 (1971), pp.\ 89--111

work page 1971
[11]

D.\ Marker, Model Theory: An Introduction, Graduate texts in mathematics, Springer, New York (2006)

work page 2006
[12]

D.\ Pincus, Zermelo-Fraenkel consistency results by Fraenkel-Mostowski methods, Journal of Symbolic Logic, Volume 37 (1972), pp.\ 721--743

work page 1972
[13]

J.\ G.\ Rosenstein, Linear Orderings, Pure and Applied Mathematics, Academic Press, New York and London, Volume 98 (1982)

work page 1982

[1] [1]

A.\ Bonato, P.\ Cameron, D.\ Delić, Tournaments and orders with the pigeonhole property, Canadian Mathematical Bulletin, Volume 43, no.\ 4 (2000), pp.\ 397--405

work page 2000

[2] [2]

P.\ J.\ Cameron, The random graph, in The Mathematics of Paul Erdős, II (2nd edition) (2013) pp.\ 353--378

work page 2013

[3] [3]

U.\ Felgner, Über das Ordnungstheorem, Zeitschrift für mathematische Logik und Grundlagen der Mathematik, Volume 17 (1971), pp.\ 257--272

work page 1971

[4] [4]

L.\ A.\ Gardiner, Infinite-exponent partition relations on the real line, arXiv:2507.12361 (2025)

work page internal anchor Pith review Pith/arXiv arXiv 2025

[5] [5]

L.\ A.\ Gardiner, J.\ Schilhan, T.\ Weinert, Infinite-exponent partition relations on higher analogues of the real line, arXiv:2605.00636 (2026)

work page internal anchor Pith review Pith/arXiv arXiv 2026

[6] [6]

T.\ J.\ Jech, The Axiom of Choice, Studies in logic and the foundations of mathematics, Volume 75, North-Holland (1973)

work page 1973

[7] [7]

P.\ Jullien, Contribution à l’étude des types d’ordres dispersés, doctoral thesis, Université de Marseille (1968)

work page 1968

[8] [8]

A.\ Karagila, J.\ Schilhan, Intermediate models and Kinna-Wagner Principles, Proceedings of the American Mathematical Society Volume 154 (2026), pp.\ 393--403

work page 2026

[9] [9]

W.\ Kinna, K.\ Wagner, Über eine Abschwächung des Auswahlpostulates, Fundamenta Mathematicae Volume 42 (1955), pp.\ 75--82

work page 1955

[10] [10]

R.\ Laver, On Fraïssé's order type conjecture, Annals of Mathematics, Volume 93, no.\ 1 (1971), pp.\ 89--111

work page 1971

[11] [11]

D.\ Marker, Model Theory: An Introduction, Graduate texts in mathematics, Springer, New York (2006)

work page 2006

[12] [12]

D.\ Pincus, Zermelo-Fraenkel consistency results by Fraenkel-Mostowski methods, Journal of Symbolic Logic, Volume 37 (1972), pp.\ 721--743

work page 1972

[13] [13]

J.\ G.\ Rosenstein, Linear Orderings, Pure and Applied Mathematics, Academic Press, New York and London, Volume 98 (1982)

work page 1982