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Martin's axiom and $\omega_1^2 \longrightarrow (\omega_1^2, 3)^2$

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Pith's one-line read Martin's Axiom at $\aleph_1$ makes every witness to $\omega_1^2\nrightarrow(\omega_1^2,3)^2$ indestructible by c.c.c. forcing, and a restricted Martin's Axiom model realizes the failure with $2^{\aleph_0}=\aleph_2$.

desk verdict A short, clean forcing result that does exactly what it claims: MA_ℵ1(σ-centered) is consistent with the failure of ω₁²→(ω₁²,3)², plus a preservation theorem under full MA_ℵ1—worth a serious referee, with only minor compression issues. read the letter →

arxiv 2608.13213 v1 pith:T54LQ7CP submitted 2026-08-13 math.LO

Mohammad Golshani This is my paper · ORCID
classification math.LO MSC 03E0203E3503E40
keywords ordinalpartitionrelationsMartin'sAxiomc.c.c.forcingσ-centeredlocallyω1-σ-linkedfinite-supportiterationnegativerelation
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper studies the ordinal partition relation $\omega_1^2\nrightarrow(\omega_1^2,3)^2$, which asserts the existence of a coloring of pairs of $\omega_1\cdot\omega_1$ with no color-0 homogeneous set of order type $\omega_1^2$ and no color-1 homogeneous triple (a triangle). Its first theorem shows that under $\mathrm{MA}_{\aleph_1}$, any such coloring in the ground model remains a witness after forcing with every c.c.c. poset, because c.c.c. forcing is locally $\omega_1$-$\sigma$-linked in that model. The second theorem is a relative-consistency result: from ZFC one can force $\mathrm{MA}_{\aleph_1}(\sigma\text{-centered})+2^{\aleph_0}=\aleph_2$ together with the failure of the partition relation. This addresses an open question in the partition calculus by showing that the positive relation does not follow from a substantial fragment of Martin's Axiom, while leaving the full question open.

What carries the argument

The load-bearing mechanism is the $\sigma$-ideal $I=\{X\subseteq\Omega:\operatorname{otp}(X)<\Omega\}$ together with the lemma that $\operatorname{otp}(X)=\Omega$ exactly when uncountably many canonical rows $R_\xi=[\omega_1\xi,\omega_1(\xi+1))$ contain uncountably many points of $X$. This lets the proof transfer a putative new 0-homogeneous set in a forcing extension back to a ground-model set of the same order type. The second piece is local $\omega_1$-$\sigma$-linkedness: every subfamily of size at most $\aleph_1$ is a countable union of linked sets. The preservation theorem shows this property is exactly what keeps the coloring intact, and the iteration theorem for $\sigma$-centered posets supplies the property at each intermediate stage of the consistency construction.

What would settle it

In the iteration of Theorem 3.1, inspect the intermediate forcing $P_\alpha$ at a limit stage $\alpha$ of cofinality $\omega_1$: the proof asserts it is $\sigma$-centered by the quoted iteration theorem, so one can look for a size-$\aleph_1$ subfamily of $P_\alpha$ that is not a countable union of linked sets; if such a family exists, the final poset is not locally $\omega_1$-$\sigma$-linked and a new 0-homogeneous set of order type $\Omega$ could appear, refuting the consistency proof.

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Extended reading notes

Core claim

The central claim is a preservation theorem. Let $\Omega=\omega_1\cdot\omega_1$ and let $c:[\Omega]^2\to 2$ be a ground-model coloring witnessing $\Omega\nrightarrow(\Omega,3)^2$. If $P$ is any forcing notion in which every subfamily of size at most $\aleph_1$ can be covered by countably many linked sets, then $c$ still witnesses the negative relation in every $P$-generic extension. The proof takes a name $\dot H$ for a 0-homogeneous set of order type $\Omega$, chooses conditions $p_x\Vdash x\in\dot H$, and uses the covering property to find a linked subfamily whose domain already has order type $\Omega$; a common extension forces all pairs in that domain to have color 0, contradicting the ground-model witness. Under $\mathrm{MA}_{\aleph_1}$, every c.c.c. forcing has the required local covering property, so every existing witness is preserved by every c.c.c. forcing. The consistency model is obtained by a finite-support iteration of $\sigma$-centered posets of length $\aleph_2$ over a model of CH carrying such a witness, with bookkeeping to force $\mathrm{MA}_{\aleph_1}(\sigma\text{-centered})$.

Load-bearing premise

The load-bearing premise is the cited iteration theorem that a finite-support iteration of $\sigma$-centered forcings of length below $\mathfrak c^+$ remains $\sigma$-centered; if that fact fails at some limit stage of cofinality $\omega_1$, the final poset in the consistency proof may lose local $\omega_1$-$\sigma$-linkedness and the ground-model coloring could be destroyed.

Editorial extensions

If this is right

  • Under $\mathrm{MA}_{\aleph_1}$, every c.c.c. forcing extension of a model containing a witness to $\omega_1^2\nrightarrow(\omega_1^2,3)^2$ still contains that witness, so c.c.c. forcing cannot be used to prove the positive relation from such a model.
  • Every $\sigma$-centered or $\sigma$-linked forcing preserves every ground-model witness to the negative relation, regardless of Martin's Axiom.
  • The model built in the paper satisfies $\mathrm{MA}_{\aleph_1}(\sigma\text{-centered})+2^{\aleph_0}=\aleph_2+\omega_1^2\nrightarrow(\omega_1^2,3)^2$, and also $\mathfrak b=\mathfrak d=\mathfrak c=\aleph_2$.
  • As a corollary, $\omega_1^2\nrightarrow(\omega_1^2,3)^2$ does not imply $\mathfrak d=\aleph_1$, so the implication from $\mathfrak d=\aleph_1$ to the negative relation does not reverse in ZFC.
  • The preservation mechanism is insensitive to the particular ground-model coloring: any witness is preserved by any poset with the local linked-cover property.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • A likely next step the paper does not take is to apply the same local-cover argument to larger finite cliques $k$, which may classify which relations $\omega_1^2\nrightarrow(\omega_1^2,k)^2$ are c.c.c.-absolute under $\mathrm{MA}_{\aleph_1}$.
  • If the full open question has a positive answer under $\mathrm{MA}_{\aleph_1}$, the paper's theorem implies that the large 0-homogeneous set would have to be created by the initial model or by non-c.c.c. means; c.c.c. forcing alone cannot create it from a ground-model witness.
  • The consistency result suggests that the negative relation is compatible with several continuum-sized cardinal characteristics being large, so the partition relation is not simply a cardinal-characteristic boundary; further iterations may show the failure persists with stronger forcing axioms than the $\sigma$-centered fragment treated here.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

0 major / 7 minor

Summary. The paper studies the Erdős–Hajnal ordinal partition relation ω_1^2 → (ω_1^2,3)^2 under Martin's axiom. It first proves a preservation theorem: if c is a coloring witnessing the negative relation, then any poset that is locally ω_1-σ-linked (every subfamily of size ≤ℵ1 is a countable union of linked sets) forces with the same c still a witness. Since MA_ℵ1 implies every c.c.c. poset is locally ω_1-σ-centered, every c.c.c. forcing preserves every existing witness in a model of MA_ℵ1 (Theorem 1.1). The second main result builds a model of MA_ℵ1(σ-centered)+2^{ℵ0}=ℵ2 plus the negative relation by a finite-support iteration of σ-centered forcings of length ω2 over GCH, starting from Hajnal's CH coloring; it preserves the coloring via the preservation theorem, and additionally satisfies b=d=c=ℵ2. The paper does not settle the original question under full MA_ℵ1, which is clearly stated.

Significance. If the results are correct, the paper makes solid progress on a long-standing problem of Erdős and Hajnal. It shows that full Martin's axiom does not trivially force the positive relation: any witness that might exist is immune to all c.c.c. forcing, a strong stability property. The consistency result shows that a natural fragment of MA_ℵ1 together with c=ℵ2 is compatible with the negative relation, and it also gives b=d=c=ℵ2. The preservation theorem is clean, elementary, and self-contained, and the local σ-linked condition is a useful new notion. Strengths include the careful forcing proof of Theorem 2.3 and the use of the ideal I to extract a ground-model homogeneous set from a generic name; the iteration part is standard but valid. The main open question under full MA_ℵ1 remains untouched, which is appropriately stated.

minor comments (7)
  1. [§2, Lemma 2.1] The lemma is stated without proof. I recommend adding a proof sketch of the equivalence and of the countable-union closure, since the σ-ideal property is used in Theorem 2.3.
  2. [§2, Definition 2.2] The term "σ-centered" is used in Fact 2.5 and Corollary 2.4 but never defined. Add the definition and note the implication centered ⇒ linked.
  3. [§2, Theorem 2.3] The proof covers only the 0-homogeneous side of the negative relation. Please add a sentence explaining that the color-1 side (no triangle) is preserved automatically because c is a ground-model coloring and any triple of ordinals is absolute.
  4. [§3, Theorem 3.1] The proof says "The standard nice-name argument then gives MA_ℵ1(σ-centered) and c=ℵ2." This is quite compressed; please specify the bookkeeping (how names for σ-centered forcings and ℵ1-sized dense families are coded by nice names) and why there are only ℵ2 many tasks to handle in an iteration of length ω2.
  5. [§3, Theorem 3.1] When applying the iteration theorem to P_α, make explicit that GCH gives c^V=ℵ1 and hence α<ω2 = c^{+V}, so the length condition is satisfied at every α, including limit stages of cofinality ω1.
  6. [§3, Theorem 3.1] "Put a Cohen forcing at cofinally many stages" should state why this is a σ-centered iterand and why it forces 2^{ℵ0}=ℵ2 in the final model.
  7. [Title and abstract] The typesetting of the arrow relation is scrambled in the title and abstract in the version I read; please ensure the final version renders ω_1^2 → (ω_1^2,3)^2 correctly.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the derivation is self-contained modulo independent cited theorems, with no fitting, self-citation, or definitional entanglement.

full rationale

The derivation chain is linear and non-circular. Theorem 1.1 is obtained by combining the preservation lemma (Theorem 2.3) with Lemma 2.6, which derives local omega1-sigma-centeredness of c.c.c. forcings under MA_aleph1 from Fact 2.5, quoted from [1]; the latter is an independent standard theorem and is not the paper's target. Theorem 2.3 proves preservation from the definition of local omega1-sigma-linkedness and Lemma 2.1, with no assumption of the conclusion. Theorem 3.1 constructs the model from GCH, Hajnal's known negative coloring, and Tall's independent iteration theorem [11] that finite-support iterations of sigma-centered forcings of length below c^+ are sigma-centered; the proof explicitly checks that each P_alpha for alpha < omega2 falls under the theorem's hypotheses (length alpha < c^+ in the ground model, iterands sigma-centered names). There is no fitted parameter renamed as a prediction, no self-citation chain, and no uniqueness claim imported from the author's prior work. The cited external theorems are used within their stated assumptions; any concern about Tall's theorem or the nice-name argument is a matter of proof detail or correctness risk, not circularity.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The central results rest on standard background theorems (Hajnal's coloring, MA consequences, Tall's iteration preservation) and on ZFC itself. No free parameters are fitted to data, and no new entities are postulated. The paper's added value is the preservation lemma, Theorem 2.3, and its application, not new axioms.

assumptions (5)
  • standard math ZFC is consistent (relative consistency statement)
    The paper proves relative consistency; all arguments are within ZFC.
  • domain assumption Hajnal's theorem: CH implies omega_1^2 does not arrow (omega_1^2,3)^2
    Used in Theorem 3.1 to fix a ground-model witness c; cited to [8].
  • standard math Fact 2.5: MA_kappa implies every c.c.c. forcing of cardinality at most kappa is sigma-centered
    Load-bearing for Lemma 2.6 and Theorem 1.1; cited to [1, Chapter 2].
  • standard math Tall's iteration theorem: a finite-support iteration of sigma-centered forcings of length strictly below c^+ (ground model c) is sigma-centered
    Load-bearing for Theorem 3.1 to show each initial segment P_alpha is sigma-centered; cited to [11].
  • standard math The bookkeeping/nice-name argument for MA_aleph_1(sigma-centered) in an iteration of length omega_2 under GCH
    Invoked as 'standard nice-name argument' in Theorem 3.1 to derive MA_aleph_1(sigma-centered) and c=aleph_2.

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Cite this review

Pith. "Pith review of Martin's axiom and $\omega_1^2 \longrightarrow (\omega_1^2, 3)^2$." pith.science (2026). https://pith.science/paper/T54LQ7CP

@misc{pith2026260813213,
  author       = {Pith},
  title        = {Pith review of: Martin's axiom and $\omega_1^2 \longrightarrow (\omega_1^2, 3)^2$},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/T54LQ7CP}},
  note         = {Machine review of arXiv:2608.13213}
}
abstract

Starting with CH and Hajnal's coloring, we show that a standard finite-support iteration of $\sigma$-centered forcing notions gives a model of $ MA_{\omega_1}(\sigma$-centered$)+2^{\aleph_0}=\aleph_2 +\omega_1^2\nrightarrow(\omega_1^2,3)^2.$ We also isolate a simple forcing-preservation principle: the same ground-model coloring remains a witness after forcing with any poset whose subfamilies of size at most $\omega_1$ are countable unions of linked sets. Under $\text{MA}_{\omega_1}$, every c.c.c. forcing has this local property, so every existing witness is preserved by every c.c.c. forcing over that model.

Discussion (0). Continue with ORCID to comment.

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Works this paper leans on

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