Open Colorings and Baumgartner's Axiom

Lorenzo Notaro

REVIEW 1 major objections 4 minor 25 references

The paper constructs a model of Martin's Axiom at ℵ1 plus the open coloring axiom OCA_T in which Baumgartner's Axiom fails, answering an open question in the negative.

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2026-08-03 12:55 UTC pith:4ZDFNFEA

load-bearing objection Real result with a real gap: answers two open questions, but the base case needs an ℵ1-dense 2-entangled set under CH. the 1 major comments →

arxiv 2601.01166 v2 pith:4ZDFNFEA submitted 2026-01-03 math.LO

Open Colorings and Baumgartner's Axiom

Lorenzo Notaro This is my paper

classification math.LO MSC 03E3503E0503E50

keywords Baumgartner's AxiomOpen Coloring AxiomMartin's Axiomentangled setsℵ1-dense sets of realsnon-reversible ordersforcing iterationsincreasing sets

verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper sets out to show that two standard axioms of set theory—Martin's Axiom at ℵ1 together with the open coloring axiom OCA_T—can hold while Baumgartner's Axiom fails. Baumgartner's Axiom says any two ℵ1-dense sets of reals are order-isomorphic, a strong rigidity statement. The construction produces a specific ℵ1-dense set of reals that is neither reversible nor increasing; the same set also makes the alternative open coloring axiom OCA_ARS false. This answers the open question negatively: adding OCA_T to MA_ℵ1 does not make the real line's dense suborders as rigid as Baumgartner's Axiom requires. The method is a finite-support forcing iteration that preserves a weakened form of entanglement through every stage.

Core claim

The paper proves the consistency of MA_ℵ1 + OCA_T together with the failure of Baumgartner's Axiom and of OCA_ARS. The witness is a nonstationarily 2-entangled sequence: an ℵ1-dense enumeration of distinct reals whose range has no injective, monotone, fixed-point-free partial map with a stationary domain. Any such sequence has a non-reversible range, which kills Baumgartner's Axiom, and a further argument shows it also defeats OCA_ARS. The technical heart is that these sequences survive the standard iteration for MA_ℵ1 + OCA_T: at each stage, a poset either preserves the sequence or is replaced by a poset that preserves the sequence and destroys the would-be destroyer's countable chain condi

What carries the argument

A nonstationarily 2-entangled sequence: an enumeration of distinct reals whose range is ℵ1-dense and such that every injective, monotone, fixed-point-free partial map on the range has nonstationary domain. Such a sequence's range is non-reversible, which is enough to refute Baumgartner's Axiom, and the sequence also refutes OCA_ARS. The proof's driving mechanism is a finite-support iteration: at countable-cofinality stages, a bookkeeping poset is either kept (if it preserves the sequence) or replaced by a poset that preserves the sequence and destroys the bookkeeping poset's countable chain condition; at uncountable-cofinality stages, clique-forcing for open graphs is refined so that the seq

Load-bearing premise

The construction needs a starting 2-entangled set of reals whose range is ℵ1-dense, but the proof only cites the continuum hypothesis for the existence of a 2-entangled set, without showing that it can be chosen to satisfy the density condition.

What would settle it

Prove or disprove: under CH there exists an ℵ1-dense 2-entangled set of reals. If the answer is no, then Definition 3's density requirement has no starting object and the iteration in Theorem 5 cannot get started; if the answer is yes, the missing density argument is the first step to supply.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

Baumgartner's Axiom is not provable from MA_ℵ1 + OCA_T; the two axioms can hold while two ℵ1-dense sets of reals are not isomorphic.
OCA_ARS is not a consequence of MA_ℵ1 + OCA_T, so the two open coloring principles remain independent even under Martin's Axiom.
There is a model of MA_ℵ1 containing an ℵ1-dense set of reals that is neither increasing nor reversible, so the 'reversible or increasing' dichotomy fails.
The obstruction to Baumgartner's Axiom in this model is not an increasing set—since OCA_T rules those out—but a non-reversible dense set.

Where Pith is reading between the lines

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

The nonstationarily 2-entangled template may extend: other axioms forced alongside MA_ℵ1 could be combined with preserved 'almost-bad' objects by replacing any spoiling poset with a spoiler-destroying poset.
A natural next question is whether the same iteration can preserve a family of nonstationarily entangled sequences, yielding many non-isomorphic ℵ1-dense order types under OCA_T rather than just one witness.
The distinction between 'increasing' and 'reversible' for dense sets is sharpened: OCA_T forbids the former but leaves the latter possible, suggesting a broader taxonomy of rigidity properties that can coexist with OCA_T.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Desk Editor's Note

Real result with a real gap: answers two open questions, but the base case needs an ℵ1-dense 2-entangled set under CH.

read the letter

Quick take: this is a genuine result. It answers Farah's question and the Marun–Shelah–Switzer question, and the technical core—nonstationarily 2-entangled sequences plus the Abraham–Shelah contradiction method—is a serious piece of forcing. The stress-test note is on target.

The new notion is a real strengthening: instead of killing every monotone self-embedding, you require the domain to be nonstationary relative to the enumeration. That gives you enough to kill BA and OCA_ARS while surviving an MA+OCA iteration. The proof is detailed and, as far as I can tell, the preservation argument holds together. The construction of Y via elementary chains and the two case analyses in Proposition 6 are intricate, and I did not find an obvious hole.

The soft spots are exactly the two flagged. First, the base case: Theorem 5 starts by letting E be any enumeration of a 2-entangled set, and the text claims any such enumeration is nonstationarily 2-entangled. That is false as stated. Definition 3 requires the range to be ℵ1-dense, and Definition 2.3 does not give you that. The stress-test construction—adding a countable set on one side—gives a 2-entangled set that is not ℵ1-dense. So the paper needs a proof that CH yields an ℵ1-dense 2-entangled set. That is very likely true, but it is not in the paper, and without it the iteration has nothing to preserve.

Second, the use of unpublished Todorcevic work for the OCA_ARS-to-σ-monotone implication. It is a reported result from the literature, so not fatal, but the paper should state it precisely or give a reference where it is proved.

Neither issue kills the main idea. This is a serious paper for set theorists working on forcing axioms and ℵ1-dense order types. It deserves a proper referee. My recommendation: send it to review, but with a clear request to fix the starting-object gap and to give a complete proof or precise citation for the Todorcevic result. After that, I expect it to be a solid publication.

Referee Report

1 major / 4 minor

Summary. The paper introduces a new combinatorial object, a nonstationarily 2-entangled sequence, whose range is an ℵ1-dense non-reversible set of reals. It proves, via a finite-support iteration with an Abraham–Shelah style explicit-contradiction mechanism, that it is consistent that MA_ℵ1 + OCA_T holds while Baumgartner's Axiom and OCA_ARS both fail. This yields Theorem 1 and Corollary 2, answering questions of Farah and of Marun, Shelah, and Switzer.

Significance. If the construction is completed, the paper settles two open questions and provides a useful new tool: a preservation-friendly strengthening of 2-entanglement that is compatible with OCA_T. The main technical work—Propositions 6 and 7 and the iteration argument in Theorem 5—is substantial and largely coherent, with the preservation argument organized into claims. The paper is clearly written and makes good use of earlier work by Todorcevic, Veličković, and Abraham–Shelah. The central missing point, however, is a load-bearing existence fact for the starting object; until that is supplied, the result is conditional.

major comments (1)

[§3, Definition 3 and proof of Theorem 5 (last paragraph)] The iteration requires a starting sequence satisfying Definition 3, in particular the range must be ℵ1-dense. The proof of Theorem 5 says only 'we let ⃗E be any enumeration of some 2-entangled set—recall that the existence of an entangled set follows from CH', and the note before Proposition 6 asserts that any enumeration of a 2-entangled set is nonstationarily 2-entangled. But §2.3 defines 2-entangled without any density condition, so a CH-produced 2-entangled set need not be ℵ1-dense. For instance, adding a countable set with no largest element below a 2-entangled set preserves 2-entanglement while destroying ℵ1-density. Thus the preservation arguments in Propositions 6 and 7 have no guaranteed ground object. This is likely fixable by explicitly constructing an ℵ1-dense 2-entangled set under CH or by otherwise proving the required starting object exists, but the paper as written does n

minor comments (4)

[§2.2 and Lemma 4] The implication OCA_ARS ⇒ σ-monotone is used in Lemma 4. It is elementary, but the paper cites only an unpublished Todorcevic result for a related equivalence. A short proof or more precise citation for the implication itself would be helpful.
[Proof of Theorem 5] Propositions 6 and 7 are stated under CH, but the proof of Theorem 5 does not explicitly justify that CH holds at every intermediate stage of the iteration. This is standard, since all iterands are ccc and of size at most ℵ1 under CH, but a one-sentence comment would improve readability.
[Definition 3] The phrase 'without fixed points' is used essentially in Proposition 7 and Claim 7.1 but is not explicitly defined for partial monotone maps. Please define it.
[§3, Proposition 6] The proof that H(Y,G) is ccc is delegated to 'minor modifications' of Todorcevic's classical argument. Since this is a central step, a more detailed citation or a sketch of the modification would be appropriate for a journal submission.

Circularity Check

0 steps flagged

No circular derivation: preservation argument reduces to Definition 3 and standard external forcing theorems; the Theorem 5 starting-object gap is an incompleteness, not a circularity.

full rationale

No circular step is load-bearing. Definition 3 introduces a genuinely new notion (nonstationarily 2-entangled sequences), and Lemma 4 derives the failures of BA and OCA_ARS directly from that definition, not from the notion being defined in terms of the conclusion. Propositions 6 and 7 are proved from the definition via standard elementary-submodel, ccc, and open-graph arguments; their external inputs are standard facts about OCA_ARS and the known OCA_T forcing from [20, 22, 24]. The author's own [7] appears only in a survey sentence about recent literature and is not used in the proof. The flagged issue in Theorem 5 is a correctness gap rather than circularity: the proof says 'we let ⃗E be any enumeration of some 2-entangled set—recall that the existence of an entangled set follows from CH', and the note before Proposition 6 says 'any enumeration of a 2-entangled set of reals is nonstationarily 2-entangled'; but Definition 3 also requires ran(⃗E) to be ℵ1-dense, while the cited 2-entangled sets from Section 2.3 are only uncountable. As written, the existence of a suitable starting object is not certified by the cited CH construction. This is an omitted or under-supplied bridge in the proof, not a reduction of the theorem to its own input; a strengthened CH construction producing an ℵ1-dense 2-entangled set would repair it without changing the derivation's structure.

Axiom & Free-Parameter Ledger

0 free parameters · 5 axioms · 1 invented entities

No free parameters or fitted constants. The proof rests on standard set-theoretic axioms and cited theorems. Two external facts deserve scrutiny: the unpublished Todorcevic σ-monotonicity result and the existence of an ℵ1-dense 2-entangled set under CH.

axioms (5)

domain assumption Con(ZF), equivalently there is a model of ZFC; the theorem is a relative consistency result.
The theorem claims consistency relative to ZF, so the background assumption of a set-theoretic universe is standard and unavoidable.
domain assumption The ground model satisfies V=L.
Used in the proof of Theorem 5 to obtain diamond on {α<ω2 : cof(α)=ω1} and to produce an entangled set. This is a standard way to build models of forcing axioms.
domain assumption Under CH there exists an ℵ1-dense 2-entangled set of reals.
Invoked in the proof of Theorem 5 via 'the existence of an entangled set follows from CH'. The paper cites [2, 21], but does not prove the ℵ1-density version needed by Definition 3.
domain assumption OCA_ARS implies that every injective f:A→R with |A|=ℵ1 is σ-monotone.
Section 2.2 reports this as an unpublished result of Todorcevic, reported in [19]. Lemma 4 uses it to derive the failure of OCA_ARS from the existence of a nonstationarily 2-entangled sequence.
standard math Todorcevic's classical argument shows H(Y,G) is ccc for the Y constructed in Proposition 6.
The proof says 'With minor modifications, the classical argument due to Todorcevic [24, Theorem 2.1] shows H(Y,G) is ccc'. This is a cited external result.

invented entities (1)

Nonstationarily 2-entangled sequence no independent evidence
purpose: A rigid ℵ1-dense set of reals that survives the MA + OCA iteration and witnesses failure of BA and OCA_ARS.
This is a new definition introduced by the paper. It is not an unexplained physical postulate; its existence in a model is the theorem being proved.

pith-pipeline@v1.3.0-alltime-deepseek · 10426 in / 33302 out tokens · 303116 ms · 2026-08-03T12:55:16.828589+00:00 · methodology

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read the original abstract

We construct a model of $\mathsf{MA_{\aleph_1}}+\mathsf{OCA}_T$ where Baumgartner's Axiom fails, settling a question of Farah. Moreover, in the same model there is an $\aleph_1$-dense set of reals which is neither reversible nor increasing, answering a question of Marun, Shelah, and Switzer.

discussion (0)

Reference graph

Works this paper leans on

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OCAand automorphisms ofP(ω)/fin

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1993

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On the consistency of some partition theorems for continuous colorings, and the structure ofℵ 1-dense real order types

Uri Abraham, Matatyahu Rubin, and Saharon Shelah. “On the consistency of some partition theorems for continuous colorings, and the structure ofℵ 1-dense real order types”. In:Ann. Pure Appl. Logic29.2 (1985), pp. 123–206.issn: 0168-0072

1985

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Martin’s axiom does not im- ply that every twoℵ1-dense sets of reals are isomorphic

Uri Abraham and Saharon Shelah. “Martin’s axiom does not im- ply that every twoℵ1-dense sets of reals are isomorphic”. In:Israel Journal of Mathematics38.1 (1981), pp. 161–176

1981

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Allℵ 1-dense sets of reals can be isomor- phic

James E. Baumgartner. “Allℵ 1-dense sets of reals can be isomor- phic”. In:Fund. Math.79.2 (1973), pp. 101–106.issn: 0016-2736, 1730-6329

1973

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Applications of the proper forcing ax- iom

James E. Baumgartner. “Applications of the proper forcing ax- iom”. In:Handbook of set-theoretic topology. North-Holland, Am- sterdam, 1984, pp. 913–959.isbn: 0-444-86580-2

1984

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Narrow Boolean algebras

Robert Bonnet and Saharon Shelah. “Narrow Boolean algebras”. In:Ann. Pure Appl. Logic28.1 (1985), pp. 1–12.issn: 0168-0072, 1873-2461

1985

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Uniform Roe coronas

Bruno M. Braga, Ilijas Farah, and Alessandro Vignati. “Uniform Roe coronas”. In:Adv. Math.389 (2021), Paper No. 107886, 35. issn: 0001-8708, 1090-2082

2021

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Rapha¨ el Carroy, Maxwell Levine, and Lorenzo Notaro.Some ques- tions on entangled linear orders. 2025. arXiv:2507.17503. REFERENCES 13

Pith/arXiv arXiv 2025

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Jorge Antonio Cruz Chapital.There may be ann-entangled set but non+ 1-entangled sets. 2025. arXiv:2509.01029

Pith/arXiv arXiv 2025

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Why Y-c.c

David Chodounsk´ y and Jindˇ rich Zapletal. “Why Y-c.c.” In:Ann. Pure Appl. Logic166.11 (2015), pp. 1123–1149.issn: 0168-0072, 1873-2461

2015

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OCA and towers inP(N)/fin

Ilijas Farah. “OCA and towers inP(N)/fin”. In:Comment. Math. Univ. Carolin.37.4 (1996), pp. 861–866.issn: 0010-2628, 1213- 7243

1996

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All automorphisms of the Calkin algebra are inner

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2011

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Almost disjoint families and the geometry of nonseparable spheres

Osvaldo Guzm´ an, Michael Hruˇ s´ ak, and Piotr Koszmider. “Almost disjoint families and the geometry of nonseparable spheres”. In: J. Funct. Anal.285.11 (2023), Paper No. 110149, 49.issn: 0022- 1236, 1096-0783

2023

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TheP-ideal dichotomy, Martin’s axiom and entangled sets

Osvaldo Guzm´ an and Stevo Todorˇ cevi´ c. “TheP-ideal dichotomy, Martin’s axiom and entangled sets”. In:Israel J. Math.263.2 (2024), pp. 909–963.issn: 0021-2172, 1565-8511

2024

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The Third Millennium Edition

Thomas Jech.Set Theory. The Third Millennium Edition. Springer Monographs in Mathematics. Springer-Verlag, Berlin, 2003.isbn: 3-540-44085-2

2003

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Carlos Martinez-Ranero and Lucas Polymeris.Entangled Suslin lines andOGA. 2025. arXiv:2512.01065

arXiv 2025

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Pedro Marun, Saharon Shelah, and Corey Bacal Switzer.Baum- gartner’s Axiom and Small Posets. 2025. arXiv:2512.21247

arXiv 2025

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Forcing axioms and coronas of C∗-algebras

Paul McKenney and Alessandro Vignati. “Forcing axioms and coronas of C∗-algebras”. In:J. Math. Log.21.2 (2021), Paper No. 2150006, 73.issn: 0219-0613, 1793-6691

2021

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A fragment of As- per´ o-Mota’s finitely proper forcing axiom and entangled sets of reals

Tadatoshi Miyamoto and Teruyuki Yorioka. “A fragment of As- per´ o-Mota’s finitely proper forcing axiom and entangled sets of reals”. In:Fund. Math.251.1 (2020), pp. 35–68.issn: 0016-2736, 1730-6329

2020

[19] [19]

Weak diamond and open colorings

Justin T. Moore. “Weak diamond and open colorings”. In:J. Math. Log.3.1 (2003), pp. 119–125.issn: 0219-0613,1793-6691

2003

[20] [20]

Some remarks on the Open Coloring Axiom

Justin T. Moore. “Some remarks on the Open Coloring Axiom”. In:Ann. Pure Appl. Logic172.5 (2021), Paper No. 102912, 6. issn: 0168-0072, 1873-2461

2021

[21] [21]

Remarks on chain conditions in products

Stevo Todorˇ cevi´ c. “Remarks on chain conditions in products”. In:Compositio Math.55.3 (1985), pp. 295–302.issn: 0010-437X, 1570-5846. 14 REFERENCES

1985

[22] [22]

Stevo Todorˇ cevi´ c.Partition problems in topology. Vol. 84. Con- temporary Mathematics. American Mathematical Society, Prov- idence, RI, 1989, pp. xii+116.isbn: 0-8218-5091-1

1989

[23] [23]

A proof of Nogura’s conjecture

Stevo Todorˇ cevi´ c. “A proof of Nogura’s conjecture”. In:Proc. Amer. Math. Soc.131.12 (2003), pp. 3919–3923.issn: 0002-9939, 1088-6826

2003

[24] [24]

Applications of the open coloring axiom

Boban Veli˘ ckovi´ c. “Applications of the open coloring axiom”. In: Set theory of the continuum (Berkeley, CA, 1989). Vol. 26. Math. Sci. Res. Inst. Publ. Springer, New York, 1992, pp. 137–154.isbn: 0-387-97874-7

1989

[25] [25]

OCAand automorphisms ofP(ω)/fin

Boban Veli˘ ckovi´ c. “OCAand automorphisms ofP(ω)/fin”. In: Topology Appl.49.1 (1993), pp. 1–13.issn: 0166-8641, 1879-3207. University of Vienna, Institute of Mathematics, Kurt G¨odel Re- search Center, Kolingasse 14-16, 1090 Vienna, Austria Email address:lorenzo.notaro@univie.ac.at

1993