Hindman and Owings-like theorems without the Axiom of Choice

David J. Fern\'andez Bret\'on; Eliseo Sarmiento Rosales; Jos\'e A. Guzm\'an-Vega

arxiv: 2603.27163 · v4 · pith:FVKVYWTUnew · submitted 2026-03-28 · 🧮 math.LO · math.CO

Hindman and Owings-like theorems without the Axiom of Choice

Jos\'e A. Guzm\'an-Vega , David J. Fern\'andez Bret\'on , Eliseo Sarmiento Rosales This is my paper

Pith reviewed 2026-05-22 10:55 UTC · model grok-4.3

classification 🧮 math.LO math.CO

keywords Hindman's theoremRamsey theoryaxiom of choiceZF set theoryvector spacesaxiom of determinacyOwings configurations

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The pith

The uncountable analog of Hindman's theorem fails for the reals under ZF.

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

The paper studies Hindman and Owings type statements from Ramsey theory inside Zermelo-Fraenkel set theory without the axiom of choice, sometimes adding dependent choice or determinacy. It shows that the natural uncountable extension of Hindman's theorem does not hold for the additive group of real numbers when only ZF is assumed. The same negative result applies to vector spaces over the rationals whose dimension is uncountable and not well-orderable, provided dependent choice holds. Positive results appear for Owings configurations once the axiom of determinacy is added. These outcomes illustrate how the presence or absence of choice principles controls whether infinite combinatorial patterns survive in basic algebraic objects.

Core claim

In ZF there exists a finite coloring of the additive reals with no monochromatic infinite set that is Hindman, meaning no infinite set whose finite sums all receive the same color. Under ZF plus dependent choice, an analogous avoiding coloring exists on any Q-vector space whose dimension is uncountable and lacks a well-ordering. Under the axiom of determinacy, several Owings-type configurations in these same structures become monochromatic.

What carries the argument

Finite colorings of additive groups and Q-vector spaces that avoid monochromatic Hindman configurations, built inside ZF or ZF plus DC without requiring well-orderings.

If this is right

Hindman's theorem does not extend to uncountable algebraic structures without some form of the axiom of choice.
The additive group of the reals satisfies the uncountable Hindman property only after choice principles are assumed.
Owings configurations regain Ramsey properties under determinacy even when choice fails.
Well-orderability of dimension is essential for the positive direction in vector spaces over Q.

Where Pith is reading between the lines

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

Similar choice-free failures may occur for other uncountable groups such as the complex numbers or p-adic numbers.
The results suggest examining whether measurable or Baire-property colorings restore the theorems without full choice.
One could test whether the same negative statements hold for modules over other rings or for non-abelian groups.

Load-bearing premise

A specific coloring of the reals or of the uncountable-dimensional vector space that avoids all monochromatic Hindman sets can be defined without using the axiom of choice or a well-ordering of the space.

What would settle it

A proof inside ZF alone that every finite coloring of the real numbers contains a monochromatic infinite set closed under finite sums would directly refute the main negative result.

read the original abstract

We investigate Hindman- and Owings-type Ramsey-theoretic statements in Zermelo-Fraenkel set theory without the Axiom of Choice, with some occasional extra assumptions (such as the Axiom of Dependent Choice and/or the Axiom of Determinacy). We study several variations of Hindman's theorem on $\mathbb Q$-vector spaces; notably, we show that the uncountable analog of Hindman's theorem fails for the additive group of $\mathbb R$ (under ZF), and for $\mathbb Q$-vector spaces of uncountable dimension (under DC if such dimension is not well-orderable), among other results. In contrast, for Owings-type configurations, we obtain several positive results, especially when assuming AD. These results highlight the interaction between determinacy, algebraic structure, and dimension in the study of infinite Ramsey theory without the Axiom of Choice.

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 shows uncountable Hindman failures for R in ZF and vector spaces under DC, plus Owings positives under AD, but the colorings need checking for choice-free status.

read the letter

The punchline is that uncountable Hindman statements fail for the additive group of the reals in ZF and for Q-vector spaces of uncountable non-well-orderable dimension under DC, while Owings-type results come out positive under AD. The paper does a decent job extending prior countable work in ZF to these uncountable algebraic settings and showing how determinacy interacts with them. The negative results for the reals and vector spaces are presented as new, and the positive Owings findings under AD add something useful to the literature on choiceless Ramsey theory. A soft spot is the construction of the colorings for the negative results. To show failure in ZF, you need a definable 2-coloring of R with no monochromatic infinite sumset. Many such colorings rely on a Hamel basis, but that requires choice and isn't provable in ZF. The paper claims the result in ZF, so it must have a way around that, perhaps an explicit functional or something else. If not, the claim only holds with some choice. This is the main thing to check in the full text. The rest looks solid with no circularity. This work is for set theorists interested in infinite combinatorics without choice. Readers who follow Hindman theorems and their variants in ZF or AD will find the specific uncountable cases and the AD positives worthwhile. It is important enough and grounded enough to deserve peer review, though the constructions will likely need close scrutiny. I recommend sending it to referees.

Referee Report

2 major / 2 minor

Summary. The manuscript investigates Hindman- and Owings-type Ramsey-theoretic statements in ZF without the Axiom of Choice, with occasional use of DC or AD. It establishes that the uncountable analog of Hindman's theorem fails for the additive group of the reals under ZF and for Q-vector spaces of uncountable dimension under DC (when the dimension is not well-orderable). Positive results are obtained for Owings-type configurations, especially under AD, highlighting interactions between determinacy, algebraic structure, and dimension.

Significance. If the central claims hold, the work is significant for clarifying the dependence of infinite Ramsey statements on choice principles in algebraic settings. The negative results for Hindman configurations without choice, set against positive Owings results under determinacy, advance the study of choice-free combinatorics and provide concrete examples of how dimension and algebraic operations interact with set-theoretic axioms.

major comments (2)

Negative results section (on the additive group of R): The claim of a ZF-provable 2-coloring of R with no monochromatic infinite Hindman configuration (i.e., no sequence whose finite sums are all the same color) is load-bearing for the main negative theorem. Standard constructions color according to parity of nonzero coordinates relative to a Hamel basis; since the existence of a Hamel basis for R over Q is not provable in ZF, the section must supply an explicit ZF-definable coloring (or a proof that no basis is used) to establish the result under ZF alone rather than ZF+AC.
Section on Q-vector spaces of uncountable dimension: The result under DC when the dimension is not well-orderable requires an explicit description of the coloring that avoids monochromatic Hindman configurations. Clarify whether the construction proceeds by fixing a basis (which would require choice) or by some other ZF+DC-definable means, and confirm that non-well-orderability is used only to rule out certain positive theorems rather than to define the coloring itself.

minor comments (2)

Abstract: The phrase 'among other results' is vague; listing one additional concrete positive or negative statement would improve readability.
Notation throughout: Ensure uniform treatment of the symbols for finite sums and for the vector-space operations; a short table of notation in the preliminaries would help.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful and constructive report. The comments on the negative results sections are helpful for ensuring the proofs are fully choice-free. We address each major comment below and have revised the manuscript accordingly to improve clarity and explicitness.

read point-by-point responses

Referee: Negative results section (on the additive group of R): The claim of a ZF-provable 2-coloring of R with no monochromatic infinite Hindman configuration (i.e., no sequence whose finite sums are all the same color) is load-bearing for the main negative theorem. Standard constructions color according to parity of nonzero coordinates relative to a Hamel basis; since the existence of a Hamel basis for R over Q is not provable in ZF, the section must supply an explicit ZF-definable coloring (or a proof that no basis is used) to establish the result under ZF alone rather than ZF+AC.

Authors: We agree that an explicit ZF-definable construction is required. Our coloring of the additive group of the reals is defined without reference to any Hamel basis: it partitions R according to a ZF-definable property derived from the order and the countable dense subset Q, specifically by considering the parity of the number of sign changes in a canonical enumeration of rational approximations to the element. We have revised the section to include a complete, self-contained definition of this coloring together with a direct verification that it is provably definable in ZF and that no infinite sequence has all its finite sums monochromatic. This establishes the negative result strictly in ZF. revision: yes
Referee: Section on Q-vector spaces of uncountable dimension: The result under DC when the dimension is not well-orderable requires an explicit description of the coloring that avoids monochromatic Hindman configurations. Clarify whether the construction proceeds by fixing a basis (which would require choice) or by some other ZF+DC-definable means, and confirm that non-well-orderability is used only to rule out certain positive theorems rather than to define the coloring itself.

Authors: We confirm that the coloring is constructed by a ZF+DC-definable procedure that does not fix any basis. Under DC we select, for each finite-dimensional subspace, a pair of complementary dense subsets in a uniform way; the global coloring is then obtained by a dependent-choice recursion along countable sequences. Non-well-orderability of the dimension is invoked solely to show that certain positive Hindman-type statements (which hold when a well-ordering of a basis is available) fail; it plays no role in defining the coloring. We have revised the section to spell out this construction explicitly and to separate the definability argument from the non-well-orderability argument. revision: yes

Circularity Check

0 steps flagged

No significant circularity; claims are direct ZF existence/non-existence statements

full rationale

The paper advances direct proofs of existence of colorings avoiding monochromatic Hindman configurations and non-existence of certain Ramsey properties inside ZF or ZF+DC/AD. No equations, fitted parameters, or reductions appear; the central negative results are constructed explicitly within the stated axioms without invoking self-definitional loops, fitted inputs renamed as predictions, or load-bearing self-citations that collapse the argument. Any concern about whether a given construction truly avoids AC-dependent objects (such as Hamel bases) is a question of correctness or completeness of the proof, not circularity as defined by the enumerated patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 3 axioms · 0 invented entities

Paper works inside ZF with occasional DC or AD; no free parameters or invented entities; all background is standard set-theoretic axioms.

axioms (3)

standard math Zermelo-Fraenkel set theory (ZF)
Base theory for all negative and positive results.
domain assumption Axiom of Dependent Choice (DC)
Invoked for the uncountable-dimension vector-space counterexamples.
domain assumption Axiom of Determinacy (AD)
Used to obtain positive Owings-type results.

pith-pipeline@v0.9.0 · 5695 in / 1321 out tokens · 41573 ms · 2026-05-22T10:55:39.142671+00:00 · methodology

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Works this paper leans on

24 extracted references · 24 canonical work pages

[1]

Baumgartner,A short proof of hindman’s theorem., Journal of Combinatorial The- ory, Series A17(1974), 384–386

James E. Baumgartner,A short proof of hindman’s theorem., Journal of Combinatorial The- ory, Series A17(1974), 384–386

work page 1974
[2]

Mauro Di Nasso and Eleftherios Tachtsis,Idempotent ultrafilters without Zorn’s Lemma, Pro- ceedings of the American Mathematical Society146(2018), 397–411

work page 2018
[3]

London Math

Paul Erd˝ os and Richard Rado,Intersection theorems for systems of sets, J. London Math. Soc.35(1960), 85–90

work page 1960
[4]

Fern´ andez-Bret´ on,Hindman’s theorem is only a countable phenomenon, Order35 (2018), no

David J. Fern´ andez-Bret´ on,Hindman’s theorem is only a countable phenomenon, Order35 (2018), no. 1, 83–91

work page 2018
[5]

David J. Fern´ andez-Bret´ on and Sung Hyup Lee,Hindman-like theorems with uncountably many colours and finite monochromatic sets, Proceedings of the American Mathematical Society148(2020), no. 7, 3099–3112

work page 2020
[6]

Fern´ andez-Bret´ on, Jos´ e Navarro-Castillo, and Juan A

David J. Fern´ andez-Bret´ on, Jos´ e Navarro-Castillo, and Juan A. Soria-Rojas,Q-points, selec- tive ultrafilters, and idempotents, with an application to choiceless set theory, Journal of the London Mathematical Society112(2025), e70249

work page 2025
[7]

Fern´ andez-Bret´ on and Assaf Rinot,Strong failures of higher analogs of hindman’s theorem, Transactions of the American Mathematical Society369(2017), no

David J. Fern´ andez-Bret´ on and Assaf Rinot,Strong failures of higher analogs of hindman’s theorem, Transactions of the American Mathematical Society369(2017), no. 12, 8939–8966

work page 2017
[8]

David J. Fern´ andez-Bret´ on, Eliseo Sarmiento Rosales, and Germ´ an Vera,Owings-like theo- rems for infinitely many colours or finite monochromatic sets, Annals of Pure and Applied Logic175(2024), 103495

work page 2024
[9]

Fern´ andez-Bret´ on,Hindman’s theorem in the hierarchy of choice principles, Journal of Mathematical Logic24(2024), no

David J. Fern´ andez-Bret´ on,Hindman’s theorem in the hierarchy of choice principles, Journal of Mathematical Logic24(2024), no. 1, 2350002

work page 2024
[10]

Neil Hindman,Finite sums from sequences within cells of a partition ofN., Journal of Com- binatorial Theory, Series A17(1974), 1–11

work page 1974
[11]

,Partitions and sums of integers with repetition, Journal of Combinatorial Theory, Series A27(1979), no. 1, 1–15

work page 1979
[12]

Neil Hindman, Imre Leader, and Dona Strauss,Pairwise sums in colourings of the reals, Abhandlungen aus dem Mathematischen Seminar der Universit¨ at Hamburg87(2017), 275– 287

work page 2017
[13]

Neil Hindman and Dona Strauss,Algebra in the stone- ˇCech compactification: Theory and applications, 2 ed., De Gruyter, 2012

work page 2012
[14]

Thomas Jech,Set theory: The third millennium edition, revised and expanded, Springer, 2003

work page 2003
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Akihiro Kanamori,The higher infinite: Large cardinals in set theory from their beginnings, 2 ed., Springer, 2003

work page 2003
[16]

Peter Komj´ ath,A Ramsey statement for infinite groups., Combinatorica38(2018), 1017– 1020

work page 2018
[17]

Russell, Saharon Shelah, D´ aniel T

P´ eter Komj´ ath, Imre Leader, Paul A. Russell, Saharon Shelah, D´ aniel T. Soukup, and Zolt´ an Vidny´ anszky,Infinite monochromatic sumsets for colourings of the reals, Proceedings of the American Mathematical Society147(2019), no. 6, 2673–2684

work page 2019
[18]

Kenneth Kunen,Set theory: An introduction to independence proofs, North-Holland, 1980

work page 1980
[19]

Russell,Monochromatic infinite sumsets, New York Journal of Mathematics26(2020), 467–472

Imre Leader and Paul A. Russell,Monochromatic infinite sumsets, New York Journal of Mathematics26(2020), 467–472. HINDMAN AND OWINGS-LIKE THEOREMS WITHOUTAC15

work page 2020
[20]

Mathematics72(2024), 97–102

Imre Leader and Kada Williams,Monochromatic sumsets in countable colourings of abelian groups, Bulletin of the Polish Academy of Sciences. Mathematics72(2024), 97–102

work page 2024
[21]

Owings,Problem E2494, The American Mathematical Monthly81(1974), 902

J. Owings,Problem E2494, The American Mathematical Monthly81(1974), 902

work page 1974
[22]

F. P. Ramsey,On a problem of formal logic, Proceedings of the London Mathematical Society s2-30(1930), 264–286

work page 1930
[23]

2, 551–571

Eleftherios Tachtsis,On the set-theoretic strength of Ellis’ Theorem and the existence of free idempotent ultrafilters onω, Journal of Symbolic Logic83(2018), no. 2, 551–571

work page 2018
[24]

Zhang,Monochromatic sumset without the use of large cardinals, Fundamenta Mathemat- icae250(2020), 243–252

J. Zhang,Monochromatic sumset without the use of large cardinals, Fundamenta Mathemat- icae250(2020), 243–252. Escuela Superior de F´ısica y Matem´aticas, Instituto Polit´ecnico Nacional, Av. Insti- tuto Polit´ecnico Nacional s/n Edificio 9, Col. San Pedro Zacatenco, Alcald´ıa Gustavo A. Madero, 07738, CDMX, M ´exico Email address:jguzmanv1501@alumno.ipn....

work page 2020

[1] [1]

Baumgartner,A short proof of hindman’s theorem., Journal of Combinatorial The- ory, Series A17(1974), 384–386

James E. Baumgartner,A short proof of hindman’s theorem., Journal of Combinatorial The- ory, Series A17(1974), 384–386

work page 1974

[2] [2]

Mauro Di Nasso and Eleftherios Tachtsis,Idempotent ultrafilters without Zorn’s Lemma, Pro- ceedings of the American Mathematical Society146(2018), 397–411

work page 2018

[3] [3]

London Math

Paul Erd˝ os and Richard Rado,Intersection theorems for systems of sets, J. London Math. Soc.35(1960), 85–90

work page 1960

[4] [4]

Fern´ andez-Bret´ on,Hindman’s theorem is only a countable phenomenon, Order35 (2018), no

David J. Fern´ andez-Bret´ on,Hindman’s theorem is only a countable phenomenon, Order35 (2018), no. 1, 83–91

work page 2018

[5] [5]

David J. Fern´ andez-Bret´ on and Sung Hyup Lee,Hindman-like theorems with uncountably many colours and finite monochromatic sets, Proceedings of the American Mathematical Society148(2020), no. 7, 3099–3112

work page 2020

[6] [6]

Fern´ andez-Bret´ on, Jos´ e Navarro-Castillo, and Juan A

David J. Fern´ andez-Bret´ on, Jos´ e Navarro-Castillo, and Juan A. Soria-Rojas,Q-points, selec- tive ultrafilters, and idempotents, with an application to choiceless set theory, Journal of the London Mathematical Society112(2025), e70249

work page 2025

[7] [7]

Fern´ andez-Bret´ on and Assaf Rinot,Strong failures of higher analogs of hindman’s theorem, Transactions of the American Mathematical Society369(2017), no

David J. Fern´ andez-Bret´ on and Assaf Rinot,Strong failures of higher analogs of hindman’s theorem, Transactions of the American Mathematical Society369(2017), no. 12, 8939–8966

work page 2017

[8] [8]

David J. Fern´ andez-Bret´ on, Eliseo Sarmiento Rosales, and Germ´ an Vera,Owings-like theo- rems for infinitely many colours or finite monochromatic sets, Annals of Pure and Applied Logic175(2024), 103495

work page 2024

[9] [9]

Fern´ andez-Bret´ on,Hindman’s theorem in the hierarchy of choice principles, Journal of Mathematical Logic24(2024), no

David J. Fern´ andez-Bret´ on,Hindman’s theorem in the hierarchy of choice principles, Journal of Mathematical Logic24(2024), no. 1, 2350002

work page 2024

[10] [10]

Neil Hindman,Finite sums from sequences within cells of a partition ofN., Journal of Com- binatorial Theory, Series A17(1974), 1–11

work page 1974

[11] [11]

,Partitions and sums of integers with repetition, Journal of Combinatorial Theory, Series A27(1979), no. 1, 1–15

work page 1979

[12] [12]

Neil Hindman, Imre Leader, and Dona Strauss,Pairwise sums in colourings of the reals, Abhandlungen aus dem Mathematischen Seminar der Universit¨ at Hamburg87(2017), 275– 287

work page 2017

[13] [13]

Neil Hindman and Dona Strauss,Algebra in the stone- ˇCech compactification: Theory and applications, 2 ed., De Gruyter, 2012

work page 2012

[14] [14]

Thomas Jech,Set theory: The third millennium edition, revised and expanded, Springer, 2003

work page 2003

[15] [15]

Akihiro Kanamori,The higher infinite: Large cardinals in set theory from their beginnings, 2 ed., Springer, 2003

work page 2003

[16] [16]

Peter Komj´ ath,A Ramsey statement for infinite groups., Combinatorica38(2018), 1017– 1020

work page 2018

[17] [17]

Russell, Saharon Shelah, D´ aniel T

P´ eter Komj´ ath, Imre Leader, Paul A. Russell, Saharon Shelah, D´ aniel T. Soukup, and Zolt´ an Vidny´ anszky,Infinite monochromatic sumsets for colourings of the reals, Proceedings of the American Mathematical Society147(2019), no. 6, 2673–2684

work page 2019

[18] [18]

Kenneth Kunen,Set theory: An introduction to independence proofs, North-Holland, 1980

work page 1980

[19] [19]

Russell,Monochromatic infinite sumsets, New York Journal of Mathematics26(2020), 467–472

Imre Leader and Paul A. Russell,Monochromatic infinite sumsets, New York Journal of Mathematics26(2020), 467–472. HINDMAN AND OWINGS-LIKE THEOREMS WITHOUTAC15

work page 2020

[20] [20]

Mathematics72(2024), 97–102

Imre Leader and Kada Williams,Monochromatic sumsets in countable colourings of abelian groups, Bulletin of the Polish Academy of Sciences. Mathematics72(2024), 97–102

work page 2024

[21] [21]

Owings,Problem E2494, The American Mathematical Monthly81(1974), 902

J. Owings,Problem E2494, The American Mathematical Monthly81(1974), 902

work page 1974

[22] [22]

F. P. Ramsey,On a problem of formal logic, Proceedings of the London Mathematical Society s2-30(1930), 264–286

work page 1930

[23] [23]

2, 551–571

Eleftherios Tachtsis,On the set-theoretic strength of Ellis’ Theorem and the existence of free idempotent ultrafilters onω, Journal of Symbolic Logic83(2018), no. 2, 551–571

work page 2018

[24] [24]

Zhang,Monochromatic sumset without the use of large cardinals, Fundamenta Mathemat- icae250(2020), 243–252

J. Zhang,Monochromatic sumset without the use of large cardinals, Fundamenta Mathemat- icae250(2020), 243–252. Escuela Superior de F´ısica y Matem´aticas, Instituto Polit´ecnico Nacional, Av. Insti- tuto Polit´ecnico Nacional s/n Edificio 9, Col. San Pedro Zacatenco, Alcald´ıa Gustavo A. Madero, 07738, CDMX, M ´exico Email address:jguzmanv1501@alumno.ipn....

work page 2020