{"id":"58148df8-5e5c-49ab-a5a2-685250c4dc99","arxiv_id":"2608.13213","paper_version":1,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":7.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"It is consistent with MA_aleph_1(sigma-centered) plus continuum aleph_2 that omega_1^2 does not arrow (omega_1^2,3)^2, and under full MA_aleph_1 every c.c.c. forcing preserves such witnesses.","lead":"This paper proves new consistency and preservation facts about a long-standing Ramsey problem on the ordinal omega_1^2. It shows a restricted form of Martin's Axiom can coexist with Hajnal's 'bad' coloring, and that under full MA_aleph_1 such colorings survive all c.c.c. forcing.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified; the cited iteration theorem is standard and the preservation argument is sound.","rationale":"The reader accepted the paper and flagged the cited iteration theorem as the weakest assumption. I agree that this is the only non-elementary external citation, but I do not believe the concern lands: the theorem is standard, the paper's use is within its hypotheses, and the rest of the preservation argument checks out. Lemma 2.1 is correct, and Theorem 2.3's argument that a locally ω1-σ-linked poset cannot add a 0-homogeneous copy of Ω is sound. Lemma 2.6 correctly derives local ω1-σ-centeredness of every c.c.c. poset from MAℵ1 via the σ-centeredness of the generated Boolean algebra. Theorem 3.1's reduction to initial segments P_α is valid because cf(ω2)>ℵ1 under GCH and each P_α is σ-centered by the iteration theorem. Thus no load-bearing flaw was identified, and the reader's verdict of ACCEPT remains appropriate.","tokens_in":3845,"tokens_out":56628,"duration_ms":524514,"concrete_test":"Verify the statement of the iteration theorem in Tall [11] and Fischer–Koelbing–Wohofsky [7] to confirm it applies to P_α for α<ω2 when the iterands are P_β-names for σ-centered forcings and the length is an ordinal of cardinality ℵ1, in particular at limit stages of cofinality ω1. If the theorem instead requires the iterands to be σ-centered in the ground model or bounds length by ℵ0, then Theorem 3.1 would need a different proof. Otherwise the central argument stands.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The only non-elementary external dependency is the cited theorem that finite-support iterations of σ-centered forcings of length strictly below c^+ are σ-centered (refs [11,7]). Since the ground model has GCH (c=ℵ1, c^+=ℵ2) and every α<ω2 has cardinality ≤ℵ1, the theorem applies to each initial segment P_α, including limit stages of cofinality ω1. The use is within its stated hypotheses: the iterands are P_β-names for σ-centered forcings and the length is strictly below c^+. No internal gap or hidden cardinality assumption is apparent. The proof of Theorem 3.1 correctly reduces local ω1-σ-centeredness of P_ω2 to boundedness of ℵ1-sized sets (using cf(ω2)>ℵ1) and to σ-centeredness of the initial segments. I therefore find no load-bearing concern that would overturn the central claim.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":3996,"tokens_out":40539,"duration_ms":338901,"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.","major_comments":[],"minor_comments":[{"comment":"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.","section":"§2, Lemma 2.1"},{"comment":"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.","section":"§2, Definition 2.2"},{"comment":"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.","section":"§2, Theorem 2.3"},{"comment":"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.","section":"§3, Theorem 3.1"},{"comment":"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.","section":"§3, Theorem 3.1"},{"comment":"\"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.","section":"§3, Theorem 3.1"},{"comment":"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.","section":"Title and abstract"}],"recommendation":"minor_revision","confidential_remarks":"The manuscript is correct as far as I can tell, and the only requests are for expansion of standard iteration arguments. It fits the scope of the journal. No concerns about novelty or attribution: Hajnal's theorem, Larson's scale result, and Tall's iteration theorem are properly cited."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper is a genuinely new consistency result, not a repackaging. It builds a model of MA_ℵ1(σ-centered) + 2^ℵ0=ℵ2 + ω₁²↛(ω₁²,3)², and isolates a sharper preservation theorem: under full MA_ℵ1, every c.c.c. forcing preserves every existing witness to the negative relation. The key idea is local ω1-σ-linkedness, and Theorem 2.3 is the load-bearing argument. The proof is short, correct, and fills a real gap in the literature.\n\nWhat is actually new: Theorem 1.1 is not in the cited literature, and Theorem 2.3 is an elegant piece of forcing. The σ-ideal I = [Ω]^<Ω is exactly the right tool, and the centered-cover argument from local ω1-σ-linkedness is clean. The reduction of c.c.c. under MA to local ω1-σ-linkedness via Fact 2.5 and Boolean subalgebras is standard but competently handled. The iteration proof of Theorem 3.1 is compressed but uses only standard facts: finite-support iterations of σ-centered forcings of length below c⁺ are σ-centered, and the nice-name argument for MA_ℵ1(σ-centered). The consistency model is new, and the added touch that b=d=c=ℵ2 follows is a nice bonus.\n\nThe main soft spot is compression. The 'standard nice-name argument' in Theorem 3.1 is invoked without proof; that is fine for the intended audience but deserves a line or two. The iteration-theorem citation (Tall; Fischer–Koelbing–Wohofsky) is load-bearing, and while the stress-test confirms it applies at limit stages of cofinality ω1 because each P_α has cardinality ≤ℵ1 < c⁺ under GCH, the paper should say explicitly why the length bound is preserved. These are minor, not fatal. Also, the paper is honest about what it does not do: it does not resolve the Erdős–Hajnal question under full MA_ℵ1; Theorem 1.2 only gives MA(σ-centered). The citation pattern is appropriate and not self-referential in any problematic way.\n\nThis deserves serious refereeing. It is a correct, clean piece of work that will be useful to anyone working on ordinal partition relations or Martin's Axiom variants. I would be happy to see it published after minor revisions filling in the iteration details. Send it to a strong referee.","headline":"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.","tokens_in":4554,"tokens_out":1635,"would_cite":true,"duration_ms":14716,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["03E02","03E35","03E40"],"pacs":[],"model":"deepseek-v4-flash","headline":"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 $2^{\\aleph_0}=\\aleph_2$.","keywords":["ordinal partition relations","Martin's Axiom","c.c.c. forcing","σ-centered forcing","locally ω1-σ-linked","finite-support iteration","negative partition relation"],"falsifier":"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.","tokens_in":3636,"feed_emoji":"","tokens_out":18315,"duration_ms":133394,"temperature":0.7,"pith_summary":"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.","feed_headline":"Martin's Axiom freezes the ω₁² Ramsey failure against c.c.c. forcing","feed_subtitle":"Once a witness to ω₁²↛(ω₁²,3)² exists, every c.c.c. forcing preserves it in models of MA.","key_machinery":"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.","core_discovery":"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 $\\mathrm{MA}_{\\aleph_1}(\\sigma\\text{-centered})$.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"It supplies the ground-model coloring satisfying the negative partition relation, the starting point of the consistency proof.","marker":"[8]"},{"why":"It provides the iteration theorem that every finite-support iteration of $\\sigma$-centered forcings of length below $\\mathfrak c^+$ is $\\sigma$-centered, the load-bearing step in showing each intermediate poset is $\\sigma$-centered.","marker":"[11]"},{"why":"It is cited alongside [11] for the same iteration fact, covering preservation of $\\sigma$-centeredness through the finite-support iteration.","marker":"[7]"},{"why":"It is used in Corollary 3.2 to show that $\\mathfrak d=\\aleph_1$ gives a witness, allowing the model to separate the negative relation from $\\mathfrak d=\\aleph_1$.","marker":"[10]"},{"why":"It gives the standard fact that under $\\mathrm{MA}_\\kappa$ every c.c.c. forcing of size at most $\\kappa$ is $\\sigma$-centered, which yields the local covering property for c.c.c. forcing in Lemma 2.6.","marker":"[1]"}],"fun_headline_variants":["MA: every c.c.c. forcing preserves the ω₁² Ramsey witness","Under MA, c.c.c. forcing cannot break ω₁² Ramsey failure","MA ensures c.c.c. forcing keeps ω₁²↛(ω₁²,3)² witnesses","c.c.c. forcing preserves Ramsey witness over models of MA","MA freezes the Ramsey witness; c.c.c. forcing can't move it"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["MA: every c.c.c. forcing preserves the ω₁² Ramsey witness","Under MA, c.c.c. forcing cannot break ω₁² Ramsey failure","MA ensures c.c.c. forcing keeps ω₁²↛(ω₁²,3)² witnesses","c.c.c. forcing preserves Ramsey witness over models of MA","MA freezes the Ramsey witness; c.c.c. forcing can't move it"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000473,"raw_usage":{"total_tokens":2364,"prompt_tokens":973,"completion_tokens":1391,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":589,"completion_tokens_details":{"reasoning_tokens":1285}},"tokens_in":589,"tokens_out":1391,"duration_ms":9766,"temperature":1.0,"reasoning_tokens":1285,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T14:16:38.191440+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Hajnal, A negative partition relation,Proc","cited_arxiv_id":null,"evidence_quote":"It supplies the ground-model coloring satisfying the negative partition relation, the starting point of the consistency proof."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"It provides the iteration theorem that every finite-support iteration of $\\sigma$-centered forcings of length below $\\mathfrak c^+$ is $\\sigma$-centered, the load-bearing step in showing each intermediate poset is $\\sigma$-centered."},{"cited_title":"Fischer, M","cited_arxiv_id":null,"evidence_quote":"It is cited alongside [11] for the same iteration fact, covering preservation of $\\sigma$-centeredness through the finite-support iteration."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"It is used in Corollary 3.2 to show that $\\mathfrak d=\\aleph_1$ gives a witness, allowing the model to separate the negative relation from $\\mathfrak d=\\aleph_1$."},{"cited_title":"Bartoszyński and H","cited_arxiv_id":null,"evidence_quote":"It gives the standard fact that under $\\mathrm{MA}_\\kappa$ every c.c.c. forcing of size at most $\\kappa$ is $\\sigma$-centered, which yields the local covering property for c.c.c. forcing in Lemma 2.6."}],"review_version":1}