arxiv: 2604.26570 · v1 · submitted 2026-04-29 · 🧮 math.LO

Recognition: unknown

Ramsey Property and Pathological Sets: Almost Disjointness, Independence and Other Maximal Objects

Jialiang He, Jintao Luo, Shuguo Zhang

Pith reviewed 2026-05-07 11:32 UTC · model grok-4.3

classification 🧮 math.LO

keywords Ramsey propertyMAD familyMathias conjecturealmost disjoint familiesmaximal independent familiesVitali setsHamel basespointclass

0 comments

The pith

Under ZF with countable choice for the reals, the Ramsey property for every set in a good pointclass rules out the existence of any maximal almost disjoint family within that pointclass.

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

The paper establishes that if a good pointclass Γ satisfies the Ramsey property for all its members, then under the axiom system ZF plus CC_R there cannot be a maximal almost disjoint family whose members all lie in Γ. This confirms a conjecture from 1977 by Mathias. The result extends to families maximal with respect to analytic ideals and shows that similar regularity assumptions eliminate maximal independent families. Under a slightly stronger choice principle, the same assumptions also exclude Vitali sets and Hamel bases from Γ.

Core claim

Under ZF + CC_R, if the Ramsey property holds for all sets in a good pointclass Γ, then Γ contains no MAD family. This also holds for I-MAD families w.r.t. analytic ideals including ED, ED_fin, and finalpha alpha for countable ordinals alpha. If any of Baire property, Lebesgue measurability or Ramsey property holds for Γ, then no maximal independent family in Γ. Under ZF + DC_R and Ramsey property for Γ, no Vitali sets and no Hamel bases in Γ.

What carries the argument

The Ramsey property for all sets in a good pointclass Γ, used together with the choice principles CC_R or DC_R to derive contradictions with the existence of various maximal combinatorial objects.

Load-bearing premise

The pointclass Γ must be good, meaning it is closed under the operations needed for the proofs and contains all analytic sets, while the choice principles CC_R or DC_R are assumed to hold.

What would settle it

Constructing under ZF + CC_R a good pointclass Γ where every set has the Ramsey property, yet Γ contains a maximal almost disjoint family, would disprove the main claim.

Figures

Figures reproduced from arXiv: 2604.26570 by Jialiang He, Jintao Luo, Shuguo Zhang.

**Figure 1.** Figure 1: illustrates this definition. 0 ∞ A0 A1 A2 A3 A4 A5 · · · Am A0(2) A3(4) Am(k) view at source ↗

**Figure 2.** Figure 2: Definition of Φ(x) for x = {0, 2, 5, 7, 8, 10, 11, · · · }. Claim 6.6. For every H ∈ [N] ω, there exist y ∈ [H] ω and A ∈ A such that Φ(y) ⊆ A. Proof of Claim 6.6. Since A is an ED-MAD family and Φ(H) ∈ ED+, by maximality there exists A ∈ A such that Φ(H) ∩ A ∈ ED+. We construct y ⊆ H recursively. For convenience, let y(−1) = −1. Assume we have already defined y(j) for j < k(k+5) 2 . We now define the nex… view at source ↗

**Figure 3.** Figure 3: is a possible illustration of Φ(x) at Ax(3n) for a single index, where we assume α = ω, γ ω n = n + 1 and dom(Ax(3n)) = N. As Ax(3n) is a ++-set, blue nodes illustrate that it is everywhere infinitely branching. Red nodes represent the maximal path chosen by the index x(3n + 1)⌢m(x, n, t). Ax(3n) dom(Ax(3n)) 0-th column 1-st column 2-nd column x(3n + 1)-th column . . . . . . · · · · · · · · · . . . . . . ·… view at source ↗

read the original abstract

We show that under $\mathsf{ZF} + \mathsf{CC}_{\mathbb R}$, if the Ramsey property holds for all sets in a good pointclass $\Gamma$, then there is no MAD family in $\Gamma$, proving a long-standing conjecture made by A.R.D.\ Mathias in 1977. This also holds for $\mathcal I$-MAD families with respect to analytic ideals $\mathcal I$ including $\mathcal{ED}$, $\mathcal{ED}_{\mathrm{fin}}$, and $\finalpha{\alpha}$ for all countable ordinals $\alpha$. Under the same assumption, we show that if any one of the Baire property, Lebesgue measurability or Ramsey property holds for all sets in $\Gamma$, then there is no maximal independent family in $\Gamma$. Under the stronger assumption $\mathsf{ZF} + \mathsf{DC}_{\mathbb R}$, we further prove that if the Ramsey property holds for all sets in $\Gamma$, then $\Gamma$ contains no Vitali sets and thus no Hamel bases.

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

This paper proves Mathias' 1977 conjecture that the Ramsey property for a good pointclass implies no MAD families under ZF + CC_R, with clean extensions to I-MAD families and other maximal objects.

read the letter

This paper settles Mathias' 1977 conjecture: under ZF plus the choice principle CC_R, if every set in a good pointclass Γ has the Ramsey property, then Γ has no MAD family. It also shows the same non-existence for I-MAD families over analytic ideals like ED and ED_fin, no maximal independent families when Baire property or Lebesgue measurability holds for Γ, and no Vitali sets under the stronger DC_R plus Ramsey property. These are direct implications from regularity to non-existence of the maximal objects. The work does well by unifying several regularity assumptions with the absence of these pathological sets in a single framework, extending known results for analytic or projective sets to arbitrary pointclasses that satisfy the Ramsey property. The statements are sharp and the choice principles are used precisely to derive contradictions with the maximality. The soft spots are limited. The notion of a good pointclass carries the main load, so the paper needs to make its closure properties explicit and check they match standard definitions in the literature. CC_R and DC_R are a bit specialized, and a short comparison to DC or other weak choice axioms would help readers see the exact strength. No obvious circularity or fitting issues appear in the claims. This is for set theorists who work on infinitary combinatorics, choiceless models, or regularity properties. Anyone tracking results on MAD families or Mathias' questions will find it useful. It deserves a serious referee because it resolves a long-standing open problem with precise statements and new corollaries. I would send it to peer review.

Referee Report

0 major / 3 minor

Summary. The manuscript proves that under ZF + CC_R, if every set in a good pointclass Γ has the Ramsey property, then Γ contains no MAD families, thereby establishing Mathias' 1977 conjecture. Parallel results are obtained for I-MAD families relative to analytic ideals (ED, ED_fin, and finalpha_alpha for countable ordinals α), for the non-existence of maximal independent families when Γ has the Baire property, Lebesgue measurability, or Ramsey property, and for the non-existence of Vitali sets (hence Hamel bases) under the stronger ZF + DC_R when Γ has the Ramsey property.

Significance. If the arguments hold, the work resolves a long-standing open question in choiceless set theory and provides a uniform framework linking regularity properties to the non-existence of several classes of maximal objects. The results strengthen the known connections between the Ramsey property and combinatorial maximality, extend to I-MAD families for concrete analytic ideals, and clarify the role of weak choice principles CC_R and DC_R. The direct implication from regularity assumptions to non-existence, without additional parameters, is a notable strength.

minor comments (3)

[Abstract and §1] The abstract and introduction would benefit from an explicit, self-contained definition or reference for the term 'good pointclass' (including its closure properties and relation to analytic sets) to aid readers unfamiliar with the precise technical requirements.
[Theorem statements in §3] Notation for the ideals (e.g., finalpha{α}) and the precise statement of CC_R should be recalled or cross-referenced in the statements of the main theorems for improved readability.
[§4 or concluding remarks] A brief comparison table or paragraph contrasting the assumptions (ZF + CC_R vs. ZF + DC_R) and the corresponding conclusions (MAD families vs. Vitali sets) would help highlight the hierarchy of results.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive and encouraging report, including the detailed summary of our results and the recommendation to accept the manuscript. We are pleased that the work is viewed as resolving Mathias' conjecture and providing a uniform framework for these questions in choiceless set theory.

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper establishes a direct logical implication in ZF set theory: under ZF + CC_R, the assumption that every set in a good pointclass Γ has the Ramsey property entails that Γ contains no MAD family (and no I-MAD families for specified analytic ideals), proving Mathias' 1977 conjecture. Parallel results link Baire property/Lebesgue measurability/Ramsey property to absence of maximal independent families, and DC_R plus Ramsey to absence of Vitali sets. These are non-existence theorems derived from regularity properties and choice principles; the definitions of Γ, the Ramsey property, MAD families, and the choice axioms are independent of the target conclusions and do not reduce to each other by construction. No equations, fitted parameters, or self-citations are invoked as load-bearing steps that collapse the result to its inputs. The derivation is self-contained as a theorem-proving argument.

Axiom & Free-Parameter Ledger

0 free parameters · 4 axioms · 0 invented entities

The central claims rest on standard ZF set theory, specific choice principles for the reals, and the assumption that a pointclass satisfies the Ramsey property; no free parameters or invented entities are introduced.

axioms (4)

standard math ZF (Zermelo-Fraenkel set theory without choice)
The ambient foundation for all statements about sets and pointclasses.
domain assumption CC_R (countable choice for subsets of the reals)
Used to derive non-existence of MAD families from the Ramsey property assumption.
domain assumption DC_R (dependent choice for the reals)
Stronger choice principle used for the Vitali set and Hamel basis results.
domain assumption Ramsey property holds for every set in the pointclass Γ
The key regularity assumption that forces non-existence of the maximal objects.

pith-pipeline@v0.9.0 · 5482 in / 1571 out tokens · 41480 ms · 2026-05-07T11:32:02.121667+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

49 extracted references · 4 canonical work pages

[1]

Maximal almost disjoint fami- lies, determinacy, and forcing.Journal of Mathematical Logic, 22(01):2150026, 2022

Karen Bakke Haga, David Schrittesser, and Asger T¨ ornquist. Maximal almost disjoint fami- lies, determinacy, and forcing.Journal of Mathematical Logic, 22(01):2150026, 2022

2022
[2]

Ak Peters Series

Tomek Bartoszy´ nski and Haim Judah.Set Theory: On the Structure of the Real Line. Ak Peters Series. CRC Press, 1995

1995
[3]

Hamel bases and well– ordering the continuum.Proceedings of the American Mathematical Society, 146(8):3565– 3573, 2018

Mariam Beriashvili, Ralf Schindler, Liuzhen Wu, and Liang Yu. Hamel bases and well– ordering the continuum.Proceedings of the American Mathematical Society, 146(8):3565– 3573, 2018

2018
[4]

Combinatorial cardinal characteristics of the continuum

Andreas Blass. Combinatorial cardinal characteristics of the continuum. InHandbook of set theory, pages 395–489. Springer, 2009

2009
[5]

Definable maximal independent families

J¨ org Brendle, Vera Fischer, and Yurii Khomskii. Definable maximal independent families. Proceedings of the American Mathematical Society, 147(8):3547–3557, 2019

2019
[6]

Combinato- rial properties of MAD families.Canadian Journal of Mathematics, pages 1–69, 2025

J¨ org Brendle, Osvaldo Guzm´ an-Gonz´ alez, Michael Hruˇ s´ ak, and Dilip Raghavan. Combinato- rial properties of MAD families.Canadian Journal of Mathematics, pages 1–69, 2025

2025
[7]

Mad families constructed from perfect almost disjoint families.The Journal of Symbolic Logic, 78(4):1164–1180, 2013

J¨ org Brendle and Yurii Khomskii. Mad families constructed from perfect almost disjoint families.The Journal of Symbolic Logic, 78(4):1164–1180, 2013

2013
[8]

Forcing indestructibility of MAD families.Annals of Pure and Applied Logic, 132(2-3):271–312, 2005

J¨ org Brendle and Shunsuke Yatabe. Forcing indestructibility of MAD families.Annals of Pure and Applied Logic, 132(2-3):271–312, 2005

2005
[9]

Birkh¨ auser Basel, 2011

Lev Bukovsk´ y.The Structure of the Real Line, volume 71 ofMonografie Matematyczne. Birkh¨ auser Basel, 2011

2011
[10]

Almost disjoint families under determinacy

William Chan, Stephen Jackson, and Nam Trang. Almost disjoint families under determinacy. Advances in Mathematics, 437:109410, 2024

2024
[11]

Partition forc- ing and independent families.The Journal of Symbolic Logic, 88(4):1590–1612, 2023

Jorge A Cruz-Chapital, Vera Fischer, Osvaldo Guzm´ an, and JaroslavˇSupina. Partition forc- ing and independent families.The Journal of Symbolic Logic, 88(4):1590–1612, 2023

2023
[12]

Strong independence and its spectrum.Advances in Math- ematics, 430:109206, 2023

Monroe Eskew and Vera Fischer. Strong independence and its spectrum.Advances in Math- ematics, 430:109206, 2023

2023
[13]

More on cardinal invariants of analytic P-ideals.Com- ment

Barnab´ as Farkas and Lajos Soukup. More on cardinal invariants of analytic P-ideals.Com- ment. Math. Univ. Carolin, 50(2):281–295, 2009

2009
[14]

The spectrum of independence.Archive for Mathematical Logic, 58(7):877–884, 2019

Vera Fischer and Saharon Shelah. The spectrum of independence.Archive for Mathematical Logic, 58(7):877–884, 2019

2019
[15]

PhD thesis, Universidad Nacional Aut¨ onoma de M¨ exico, 2017

Osvaldo Guzm´ an.P-points, mad families and cardinal invariants. PhD thesis, Universidad Nacional Aut¨ onoma de M¨ exico, 2017

2017
[16]

Harrington and Alexander S

Leo A. Harrington and Alexander S. Kechris. On the determinacy of games on ordinals. Annals of Mathematical Logic, 20:109–154, 1981

1981
[17]

Madness and regularity properties.arXiv preprint arXiv:1704.08327, 2017

Haim Horowitz and Saharon Shelah. Madness and regularity properties.arXiv preprint arXiv:1704.08327, 2017

work page arXiv 2017
[18]

On the non-existence of mad families.Archive for Math- ematical Logic, 58(3):325–338, 2019

Haim Horowitz and Saharon Shelah. On the non-existence of mad families.Archive for Math- ematical Logic, 58(3):325–338, 2019

2019
[19]

A Borel maximal eventually different family.Annals of Pure and Applied Logic, 175(1, Part B):103334, 2024

Haim Horowitz and Saharon Shelah. A Borel maximal eventually different family.Annals of Pure and Applied Logic, 175(1, Part B):103334, 2024

2024
[20]

A borel maximal cofinitary group.The Journal of Sym- bolic Logic, 90(2):808–821, 2025

Haim Horowitz and Saharon Shelah. A borel maximal cofinitary group.The Journal of Sym- bolic Logic, 90(2):808–821, 2025

2025
[21]

Determinacy and regularity properties for idealized forcings.Mathematical Logic Quarterly, 68(3):310–317, 2022

Daisuke Ikegami. Determinacy and regularity properties for idealized forcings.Mathematical Logic Quarterly, 68(3):310–317, 2022

2022
[22]

F.B. Jones. Measure and other properties of a Hamel basis.Bulletin of the American Math- ematical Society, 48(6):472–481, 1942

1942
[23]

Products of filters.Commentationes Mathematicae Universitatis Caroli- nae, 9(1):173–189, 1968

Miroslav Katˇ etov. Products of filters.Commentationes Mathematicae Universitatis Caroli- nae, 9(1):173–189, 1968. 30 JIALIANG HE, JINTAO LUO ∗, AND SHUGUO ZHANG

1968
[24]

Kechris.Classical Descriptive Set Theory, volume 156

Alexander S. Kechris.Classical Descriptive Set Theory, volume 156. Springer Science & Business Media, 2012

2012
[25]

Kechris, Robert M

Alexander S. Kechris, Robert M. Solovay, and John R. Steel. The axiom of determinacy and the prewellordering property. InCabal Seminar 77–79: Proceedings, Caltech-UCLA Logic Seminar 1977–79, pages 101–125. Springer, 2006

1977
[26]

PhD thesis, University of Amsterdam, 2012

Yurii Khomskii.Regularity Properties and Definability in the Real Number Continuum: ideal- ized forcing, polarized partitions, Hausdorff gaps and mad families in the projective hierarchy. PhD thesis, University of Amsterdam, 2012. ILLC Dissertation Series DS-2012-04

2012
[27]

Studies in Logic and the Foundations of Mathematics

Kenneth Kunen.Set Theory: An Introduction to Independence Proofs. Studies in Logic and the Foundations of Mathematics. North-Holland Publishing Company, 1980

1980
[28]

Larson.Extensions of the Axiom of Determinacy, volume 78

Paul B. Larson.Extensions of the Axiom of Determinacy, volume 78. American Mathematical Society, 2023

2023
[29]

Larson and Jindˇ rich Zapletal.Geometric set theory, volume 248

Paul B. Larson and Jindˇ rich Zapletal.Geometric set theory, volume 248. American Mathe- matical Society, 2020

2020
[30]

Adrian R. D. Mathias.On a generalization of Ramsey’s theorem. PhD thesis, University of Cambridge, 1969

1969
[31]

Adrian R. D. Mathias. A remark on rare filters. InInfinite and Finite Sets: To Paul Erd¨ os on His 60th Birthday, volume 3 ofBolyai J´ anos Matematikai T¨ arsulat: Colloquia mathematica Societatis J´ anos Bolyai, pages 1095–1097. North-Holland Publishing Company, 1975

1975
[32]

Adrian R. D. Mathias. Happy families.Annals of Mathematical logic, 12(1):59–111, 1977

1977
[33]

PhD thesis, Department of Mathemat- ical Sciences, Faculty of Science, University of Copenhagen, 2023

Severin Mejak.Definability of maximal discrete sets. PhD thesis, Department of Mathemat- ical Sciences, Faculty of Science, University of Copenhagen, 2023

2023
[34]

Definability of maximal cofinitary groups

Severin Mejak and David Schrittesser. Definability of maximal cofinitary groups. arXiv:2212.05318, 2022

work page arXiv 2022
[35]

Arnold W. Miller. Infinite combinatorics and definability.Annals of Pure and Applied Logic, 41(2):179–203, 1989

1989
[36]

Moschovakis.Descriptive set theory, volume 155

Yiannis N. Moschovakis.Descriptive set theory, volume 155. American Mathematical Society, 2025

2025
[37]

Happy and mad families inL(R).The Journal of Symbolic Logic, 83(2):572–597, 2018

Itay Neeman and Zach Norwood. Happy and mad families inL(R).The Journal of Symbolic Logic, 83(2):572–597, 2018

2018
[38]

Compactness of maximal eventually different families.Bulletin of the London Mathematical Society, 50(2):340–348, 2018

David Schrittesser. Compactness of maximal eventually different families.Bulletin of the London Mathematical Society, 50(2):340–348, 2018

2018
[39]

Maximal discrete sets.RIMS Kˆ okyˆ uroku, 2164:64–84, 2020

David Schrittesser. Maximal discrete sets.RIMS Kˆ okyˆ uroku, 2164:64–84, 2020

2020
[40]

Definable maximal discrete sets in forcing extensions

David Schrittesser and Asger T¨ ornquist. Definable maximal discrete sets in forcing extensions. Mathematical Research Letters, 25(5):1591–1612, 2019

2019
[41]

The Ramsey property implies no mad families

David Schrittesser and Asger T¨ ornquist. The Ramsey property implies no mad families. Proceedings of the National Academy of Sciences, 116(38):18883–18887, 2019

2019
[42]

The Ramsey property and higher dimensional mad families.arXiv:2003.10944, 2020

David Schrittesser and Asger T¨ ornquist. The Ramsey property and higher dimensional mad families.arXiv:2003.10944, 2020

work page arXiv 2003
[43]

The happy coexistence of mad families and Laver measurability.arXiv:2510.21374, 2025

David Schrittesser and Asger T¨ ornquist. The happy coexistence of mad families and Laver measurability.arXiv:2510.21374, 2025

work page arXiv 2025
[44]

Sur la question de la mesurabilit´ e de la base de M

Wac law Sierpi´ nski. Sur la question de la mesurabilit´ e de la base de M. Hamel.Fundamenta Mathematicae, 1(1):105–111, 1920

1920
[45]

Robert M. Solovay. A model of set-theory in which every set of reals is Lebesgue measurable. Annals of Mathematics, 92(1):1–56, 1970

1970
[46]

Compacts de fonctions mesurables et filtres non mesurables.Studia Math- ematica, 67(1):13–43, 1980

Michel Talagrand. Compacts de fonctions mesurables et filtres non mesurables.Studia Math- ematica, 67(1):13–43, 1980

1980
[47]

Σ1 2 and Π1 1 mad families.The Journal of Symbolic Logic, 78(4):1181–1182, 2009

Asger T¨ ornquist. Σ1 2 and Π1 1 mad families.The Journal of Symbolic Logic, 78(4):1181–1182, 2009

2009
[48]

Definability and almost disjoint families.Advances in Mathematics, 330:61– 73, 2018

Asger T¨ ornquist. Definability and almost disjoint families.Advances in Mathematics, 330:61– 73, 2018

2018
[49]

Transfinite inductions producing coanalytic sets.Fundamenta Mathe- maticae, 224(2):155–174, 2014

Zolt´ an Vidny´ anszky. Transfinite inductions producing coanalytic sets.Fundamenta Mathe- maticae, 224(2):155–174, 2014. RAMSEY PROPERTY AND PATHOLOGICAL SETS 31 College of Mathematics, Sichuan University, Chengdu, Sichuan, 610064 China; Email address:jialianghe@scu.edu.cn College of Mathematics, Sichuan University, Chengdu, Sichuan, 610064 China; Email ...

2014