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Subject and independence

Role. The reviewer worked in a fresh context from the commissioning assignment alone, took no part in writing the section, the lead, the Tools section or the reconstruction pages, and had no contact with their author or with the reviewers of the other sections. The charge was refutation. No tier is assigned and no status changes.

Subject. The source as it stood on 2026-09-28T13:50:48Z (called "the commit" below). The section "## Problem 597: graph targets" of wiki/research/erdos_1171/connections.md, lines 318--409 (the H2 at line 318 up to the line before the next H2 at line 410), and the lead page wiki/research/leads/graph_targets_under_martin_axiom/_index.md, read whole. Line numbers below refer to these two files at this commit.

Reading and depth.

• The section, every sentence, with each deduction re-derived.
• The page's Tools section (lines 32--124) as input: the interfaces of (T1), (T2), (T3) and (T4) were confirmed at each use, and the proof of (T4) was re-derived because (T3), which the section uses, is stated as its constant case. (T5) is not used by the section and was not checked.
• The reconstruction pages lemma_2_1_reconstruction.md and theorem_3_1_reconstruction.md (premises), read whole, checked for the hypotheses under which the section uses them.
• The statement blocks of the problem pages E0597, E1171 and E0924.
• Library material where a source theorem is invoked: the cards and result pages of Baumgartner (1989), Baumgartner and Hajnal (1987), Folkman (1970), Chen, Garti and Weinert (2018), Gao (2026) and Komjáth (2025); the held PDFs of Chen, Garti and Weinert (p. 5 page image and text layer, and the first page for its identity), Folkman (printed p. 19 text layer, printed p. 20 page image) and Komjáth (printed pp. 425--426 and 442--443 page images).
• docs/verification.md (Whole-claim report, Audit checklist, Grading and claim standing), docs/approach_vetting.md, docs/evidence.md (Source fidelity) and docs/math_authoring.md.

Exposures. (1) The three problem pages have no separate Statement heading; the block read for each ran from its H1 to its "Current assessment" heading and so included the page's Status line and Source paragraph. For E1171 that paragraph quotes the catalog's status wording and mentions a discussion thread and a proof-claim tab; for E0924 the Status paragraph is where the Nešetřil--Rödl theorem is recorded, and it was read for that record. (2) A heading listing showed that E0597 carries "Current assessment", "Progress" and "Known Results" sections; their contents were not read. (3) A search for the quoted sentences ran over the whole connections page and returned four one-line fragments of other sections (lines 218, 287, 452 and 542); nothing else of those sections was read. (4) The library cards were read whole, including their read-status and standing lines. (5) A working-tree status listing showed the file names of parallel reviews of other sections and folders; none was opened. Nothing under any evidence/ folder, no folder _index.md, no existing review or grade, and no web content was read; the catalog page for Problem 597 was not consulted, so the section's three catalog attributions are recorded here as unverified.

Restatement

Conventions (S1, not a claim). Ordinals are von Neumann ordinals; $[X]^2$ is the set of two-element subsets. For an ordinal $\gamma$, an ordinal $\beta$ and targets $H_1,\ldots,H_k$, each an order type or a graph, $\gamma\to(\beta,H_1,\ldots,H_k)^2_{k+1}$ means: every $c:[\gamma]^2\to\{0,\ldots,k\}$ has $X\subseteq\gamma$ with $\operatorname{otp}(X)=\beta$ and $c\equiv0$ on $[X]^2$, or for some $i$ a copy of $H_i$ in color $i$, a copy being a subset of type $H_i$ for an order type and, for a graph, an injection of the vertices of $H_i$ into $\gamma$ under which every edge of $H_i$ goes to a pair of color $i$ (subgraph copies, not induced). For $k=1$ and a graph $G$ this says: every graph on $\gamma$ has an independent set of type $\beta$ or contains $G$ as a subgraph. (Q1) asks whether $\omega_1^2\to(\omega_1\omega,G)^2$ for every graph $G$ on at most $\aleph_1$ vertices with no $K_4$ and no $K_{\aleph_0,\aleph_0}$ subgraph; (Q2) asks the same for finite $G$, for which the $K_{\aleph_0,\aleph_0}$ exclusion is vacuous. The problem page carries status open. Two conventions bear on everything below: a finite $n$-element set homogeneous in color $1$ is the same object as a copy of $K_n$ in color $1$, and a positive relation on an ordinal $\gamma$ transfers to every ordinal $\gamma'\ge\gamma$ by restriction of the coloring (T2), while a negative relation transfers only downward.

• S2. Assume $\mathrm{MA}_{\aleph_1}$ and the imported theorem that $\omega_1\omega\to(\omega_1\omega,n)^2$ for every finite $n$. Then for every finite graph $G$, $\omega_1\omega\to(\omega_1\omega,G)^2$.
• S3. Under the same hypotheses, for every finite graph $G$, $\omega_1^2\to(\omega_1\omega,G)^2$. Hence in every model of $\mathrm{MA}_{\aleph_1}$ the answer to (Q2) is yes for every finite $G$, $K_4$-free or not; with the relative consistency of $\mathrm{MA}_{\aleph_1}$ (Theorem C), ZFC does not refute (Q2). The color-merging lemma is not used.
• S4. Under the same hypotheses, for every finite $k\ge1$ and finite graphs $G_1,\ldots,G_k$, $\omega_1^2\to(\omega_1\omega,G_1,\ldots,G_k)^2_{k+1}$, with color $i$ carrying $G_i$; once by (T3) at $\alpha=\omega_1\omega$ followed by restriction, and once from the imported theorem at $n$ equal to the multicolor Ramsey number of the cliques on the $G_i$, followed by the finite Ramsey theorem and restriction.
• S5. Under any hypothesis that gives $\omega_1\omega\to(\omega_1\omega,G_i)^2$ for $i=1,\ldots,k$, with the $G_i$ graphs of any size, $\omega_1^2\to(\omega_1\omega,G_1,\ldots,G_k)^2_{k+1}$; the finite Ramsey bridge of S4 does not give this when some $G_i$ is infinite.
• S6 (obstruction). The mechanism of the folder, Lemma 2.1 with its general form (T4) and the transfers (T2), does not produce the two-color relation $\omega_1^2\to(\omega_1\omega,G)^2$ from any input weaker than that relation: (T4) with one link is its own hypothesis, and any relation $\gamma\to(\beta,G)^2$ with $\gamma\le\omega_1^2$ and $\beta\ge\omega_1\omega$ that (T2) could feed in already implies the target.
• S7. For every finite $k\ge1$ there is a finite $K_4$-free graph $F_k$ such that every coloring of its edges with $k$ colors has a monochromatic triangle: for $k=2$ by Folkman's Theorem 1, for every $k$ by the theorem of Nešetřil and Rödl.
• S8. For every finite $k\ge1$ and every such $F_k$: if $\omega_1^2\to(\omega_1\omega,F_k)^2$, then $\omega_1^2\to(\omega_1\omega,3,\ldots,3)^2_{k+1}$ with $k$ triangle targets. Consequently, if ZFC proves (Q2) at every $F_k$, ZFC proves the relation of Problem 1171 for every $k$; every model of $\omega_1^2\not\to(\omega_1\omega,3,\ldots,3)^2_{k+1}$ is a model in which (Q2) fails at the $K_4$-free finite graph $F_k$; and (Q2) at $F_3$ implies the case $k=3$ of Problem 1171.
• S9. In the converse direction, the relation of Problem 1171 yields a two-color relation $\omega_1^2\to(\omega_1\omega,G)^2$ only for graphs $G$ that are subgraphs of a triangle. (Read as: by leaving colors unused and weakening the target under (T2); see F4 for the stronger reading.)
• S10. In any model, a positive answer to (Q1) implies that every graph on $\omega_1^2$ with no independent set of type $\omega_1\omega$ contains every admissible $G$ as a subgraph; the negative relation $\omega_1^2\not\to(\omega_1\omega,K_{\aleph_0,\aleph_0})^2$ attributed to Baumgartner (hypothesis unrecorded) is exactly the failure of that condition at $K_{\aleph_0,\aleph_0}$.
• S11. The route of S2--S3 reaches (Q1) only for those infinite admissible $G$ with $\omega_1\omega\to(\omega_1\omega,G)^2$ under $\mathrm{MA}_{\aleph_1}$; the imported theorem is stated for finite $n$ only, and nothing held in the corpus decides any infinite case.
• S12. Investigation (ii) of the section and step 2 of the lead. (a) If $\mathrm{MA}_{\aleph_1}$ gives $\omega_1\omega\to(\omega_1\omega,G)^2$ for an admissible $G$, then (Q1) at $G$ is not disprovable in ZFC. (b) If the extension fails for some admissible $G$, "a counterexample $G$ under $\mathrm{MA}_{\aleph_1}$", then (Q1) has no positive answer in ZFC and $G$ is a third necessary exclusion.
• S13. Investigation (iii) of the section and step 3 of the lead. A positive answer to (Q2) at $F_3$ settles the case $k=3$ of Problem 1171; a consistent negative answer to that case "refutes (Q2) at $F_3$".
• S14 (provenance and standing sentences, not mathematics). The imported theorem is taken "from the zbMATH review and one refereed restatement"; the lead reads copies as subgraph copies, "the convention of the lemma page and of Komjáth's survey"; the lead cites a CH obstruction recorded in the first section of the connections page, which is outside this review's read set and is not assessed here.

Checklist

• Quantifiers and scope. S2--S8 carry their quantifiers correctly: the universal over finite $G$ in S3 holds in one model at a time, so "not disprovable" is the right modality; S8 is stated per $k$ with any choice of $F_k$. One defect: S12(b) moves from a failure on the initial segment $\omega_1\omega$ to a conclusion about $\omega_1^2$, a change of the ordinal in the relation with no bridge (F1).
• Circularity. None. S6 observes that (T4) at one link is the target itself; that is an obstruction statement, not a circular use.
• Model and convention changes. The subgraph reading of copies is declared and used consistently in S2, S8 and S10. Two silent identifications are harmless but should be explicit: the ordinal target $n$ read as the graph $K_n$ in S2, and (T1)'s "all of whose pairs" read as "all of whose edges" for graph targets (F6). The lead misattributes the convention to the lemma reconstruction page (F5).
• Finite and statistical overreach. None. The finite Ramsey theorem and the existence of $F_k$ are used for their exact finite content.
• Uniformity. The Ramsey number in S4 depends on $G_1,\ldots,G_k$ and the graph $F_k$ depends on $k$; both dependences are stated. No constant is claimed uniform.
• Extremal conclusions. "At least as strong" in S8 is an implication and holds. "Only for subgraphs of a triangle" in S9 is exact for the weakening route but, read as an independence statement, is unsupported (F4).
• Consequences and composition. Every "hence" was checked separately. S3 inherits the unproved premise (the imported theorem at full strength) and says so; S12(b) does not follow from its hypothesis (F1); S13's "refutes" overstates a consistency conclusion (F2).
• Computation. Inapplicable: no computation is used or claimed.
• Reproduction. Inapplicable: no rerun command or coverage claim.
• Source and verdict fidelity. The restatement of the imported theorem was checked on the page image of Chen, Garti and Weinert (p. 5): it reads that ZFC with $\mathrm{MA}_{\aleph_1}$ implies $\omega_1\omega\to(\omega_1\omega,n)^2$ for all natural numbers $n$, as the section says. Folkman's Theorem 1 and the editor's footnote were checked on the page images. Komjáth's record of the Baumgartner--Hajnal relations and of the Nešetřil--Rödl theorem was checked on the page images. The word "refereed" for the restatement is not supported by the held artifact, an arXiv preprint, or by its library card (F3).

Weakest steps

W1 (S2--S3): from the ordinal target $n$ to a graph target, and up to $\omega_1^2$. Let $G$ be a finite graph with vertex set of size $n$; if $n=0$ or $n=1$ the copy is trivial, so let $n\ge2$. Let $c:[\omega_1\omega]^2\to\{0,1\}$. The imported theorem gives $X\subseteq\omega_1\omega$ of type $\omega_1\omega$ with $c\equiv0$ on $[X]^2$, or an $n$-element $Y$ with $c\equiv1$ on $[Y]^2$. In the second case any bijection from the vertices of $G$ onto $Y$ sends every edge of $G$ to a pair of color $1$, so it is a copy of $G$ in color $1$; non-edges of $G$ impose no condition under the subgraph reading. This is S2. Now let $c:[\omega_1^2]^2\to\{0,1\}$. Since $\omega_1\omega<\omega_1^2$, the ordinal $\omega_1\omega$ is the set of ordinals below it, a subset of $\omega_1^2$, and $c\upharpoonright[\omega_1\omega]^2$ is a two-coloring to which S2 applies. A set $X\subseteq\omega_1\omega$ of type $\omega_1\omega$ has the same order as a subset of $\omega_1^2$ (membership in both), so its type is unchanged, and a copy of $G$ inside $\omega_1\omega$ is a copy inside $\omega_1^2$ with the same colors. This is S3; nothing beyond $\omega_1\omega\subseteq\omega_1^2$ is used. The consistency step is Theorem C: a model of $\mathrm{MA}_{\aleph_1}$ satisfies (Q2) for every finite $G$ simultaneously, so ZFC, if consistent, cannot prove the negation of (Q2). Composition: S3 is conditional on the imported theorem at full strength; if only $n=3$ were available, S2 would cover exactly the subgraphs of a triangle, as the lead's provisional premise says.

W2 (S8): the Folkman merge. Fix $k\ge1$ and $F_k$ as in S7, and assume $\omega_1^2\to(\omega_1\omega,F_k)^2$. Let $c:[\omega_1^2]^2\to\{0,\ldots,k\}$ and put $c'(p)=0$ if $c(p)=0$ and $c'(p)=1$ otherwise. The assumed relation applied to $c'$ gives either $X$ of type $\omega_1\omega$ with $c'\equiv0$, hence $c\equiv0$, on $[X]^2$, which is the color-$0$ alternative of the target relation; or an injection $\varphi$ of the vertices of $F_k$ into $\omega_1^2$ sending each edge $e$ to a pair $\varphi(e)$ with $c(\varphi(e))\in\{1,\ldots,k\}$. Then $e\mapsto c(\varphi(e))$ is a $k$-coloring of the edges of $F_k$, so some triangle $\{u,v,w\}$ of $F_k$ has its three edges of one color $j\ge1$; the three points $\varphi(u),\varphi(v),\varphi(w)$ are distinct and all three pairs among them have $c$-color $j$, a triangle of color $j$. The merge never uses that $F_k$ is $K_4$-free; that property only places $F_k$ inside the class (Q2) asks about, which is what the consequences need. The contrapositive gives the model statement, and $k=3$ gives the last clause.

W3 (S4--S5): (T4) with graph targets, re-derived. Let $\gamma_{j-1}\to(\gamma_j,H_j)^2$ for $j=1,\ldots,k$, with the $H_j$ arbitrary graphs or order types. Claim: $\gamma_0\to(\gamma_k,H_1,\ldots,H_k)^2_{k+1}$. For $k=1$ this is the hypothesis. For the step, let $c$ use the colors $0,\ldots,k+1$ and let $c'$ send $k+1$ to $0$ and fix the other colors. The case $k$ gives a copy of some $H_j$ ($j\le k$) in $c'$-color $j$, which is $c$-color $j$ since $c'=c$ off $\{0,k+1\}$; or $Y\subseteq\gamma_0$ of type $\gamma_k$ with $c[\,[Y]^2\,]\subseteq\{0,k+1\}$. In the latter case transport $\gamma_k\to(\gamma_{k+1},H_{k+1})^2$ to $Y$ along the order isomorphism $\pi:\gamma_k\to Y$ (pull the coloring back, take the homogeneous set or copy in $\gamma_k$, push it forward; $\pi$ is injective, preserves order and carries pairs to pairs, so types and copies are preserved) and apply it to the two-coloring of $[Y]^2$ in which $k+1$ is read as $1$. The result is a subset of $Y$ of type $\gamma_{k+1}$ of $c$-color $0$ or a copy of $H_{k+1}$ of $c$-color $k+1$. No property of the targets is used, so the constant case (T3) holds for graphs of any size, and with $\alpha=\omega_1\omega$ and the restriction of W1 it gives S4 and S5. The Ramsey alternative in S4: with $n_i$ the number of vertices of $G_i$ and $n=R(n_1,\ldots,n_k)$, merge colors $1,\ldots,k$, obtain an $n$-element set in the merged color, split it back and find an $n_i$-clique of color $i$, which contains a copy of $G_i$. For infinite $G_i$ no finite $n$ contains a copy, so that bridge stops, as S5 says; the induction bridge does not.

Strongest attack

• S2, S3. Attack 1: the direction of transfer. Positive relations restrict upward (W1); the attack would need a downward use, and there is none. Attack 2: the copy convention. Under induced copies S2 fails already for $G$ the graph on two vertices with no edge, since an $n$-element set of color $1$ contains no induced non-edge; the section and lead declare subgraph copies, which the problem's exclusion of $K_{\aleph_0,\aleph_0}$ as a subgraph also presupposes, so the attack hits a convention, not the argument. Attack 3: a finite $G$ with $\omega_1^2\not\to(\omega_1\omega,G)^2$ under $\mathrm{MA}_{\aleph_1}$. Impossible given the imported theorem, since $G\subseteq K_n$; the only exposure is the theorem's full strength, which the section discloses as a premise. Attack 4: the CH negative relation for $K_4$ against S3's "whether or not $G$ is $K_4$-free". No conflict: the two hold in different models, and the held restatement (Chen, Garti and Weinert, p. 5) draws the same conclusion, that the negative relation needs a hypothesis. The attacks failed.
• S4, S5. Attack: a hidden use of finiteness in (T3). The re-derivation W3 shows none. Attack on the Ramsey alternative: the multicolor Ramsey number must be taken for the cliques $K_{n_i}$, not for the $G_i$ themselves, which is what the sentence says. Failed.
• S6. Attack: exhibit an input of the mechanism strictly weaker than (Q2) at $G$ that still yields it. Every input the tools accept is a two-color relation $\gamma\to(\beta,G)^2$ that (T2) turns into the target when $\gamma\le\omega_1^2$ and $\beta\ge\omega_1\omega$, and the induction only adds colors. Failed; the statement is modest and exact.
• S7. Checked against Folkman's Theorem 1 on the page image ($f(3,3)=3$, so a graph of clique number $3$, hence $K_4$-free, lies in $\Gamma(3,3)$) and against Komjáth's record of the Nešetřil--Rödl theorem (for $Y$ finite without $K_{k+1}$ and $r<\omega$, a finite $X$ omitting $K_{k+1}$ in which every $r$-coloring of the edges has a monochromatic induced copy of $Y$; take $Y=K_3$). The second is second-hand, as the section and Problem 924 say. Failed.
• S8. Attack: the monochromatic triangle of $F_k$ might not be a triangle of $\omega_1^2$ because non-edges of $F_k$ are uncolored. A triangle of $F_k$ consists of three edges, all of which are colored, so the image is a triangle in the ordinal (W2). Attack: the merged relation is the wrong instance of (Q2) because $F_k$ might contain $K_{\aleph_0,\aleph_0}$; it is finite. Failed.
• S9. Attack on the literal reading: the sentence "yields (Q2) only for subgraphs of a triangle" asserts, for every other $G$, that the relation of Problem 1171 does not give (Q2) at $G$. No argument is offered for that, and none is available here: it would need, for each such $G$, a model of the Problem 1171 relations in which (Q2) fails at $G$, and for $K_4$-free $G$ such as $C_4$ nothing held says whether ZFC proves (Q2) at $G$ outright. Under the route reading the statement is exact. The refutation of the literal reading is not a counterexample but the absence of any proof; filed as F4.
• S10. Attack: "necessary" hides a stronger claim. The condition is in fact equivalent to (Q1) in the model, by unfolding the relation, so "necessary" is true and weaker than what holds. The identification of the $K_{\aleph_0,\aleph_0}$ exclusion with a failure of the condition is exactly the imported negative relation, whose hypothesis the section correctly leaves open. Failed.
• S11. Attack: find a held result deciding an infinite case. None of the held or cited material states a relation $\omega_1\omega\to(\omega_1\omega,G)^2$ or $\omega_1^2\to(\omega_1\omega,G)^2$ for an infinite $G$. Failed as a statement about the corpus; see F7 for what the sentence must not be read as.
• S12. (a) holds by S3's argument: a positive relation on $\omega_1\omega$ restricts upward. (b) is where the attack succeeded. The hypothesis, read in the sentence's own context ("whether the proof of $\omega_1\omega\to(\omega_1\omega,n)^2$ extends"), is $\omega_1\omega\not\to(\omega_1\omega,G)^2$ in a model of $\mathrm{MA}_{\aleph_1}$; the conclusion needs $\omega_1^2\not\to(\omega_1\omega,G)^2$ in some model. A negative relation does not pass from $\omega_1\omega$ to $\omega_1^2$: the witness is $G=K_3$ under CH, where $\omega_1\omega\not\to(\omega_1\omega,3)^2$ (Erdős and Hajnal, as the theorem page and the Baumgartner card record from the review) while $\omega_1^2\to(\omega_1\omega,3)^2$ is a theorem of ZFC (the ZFC relation $\omega_1^2\to(\omega_1\omega,3,3)^2$ with one color unused; Komjáth p. 425 attributes the two-color instance to Erdős and Hajnal). The witness is not an instance of the sentence's exact hypothesis, which names $\mathrm{MA}_{\aleph_1}$, so the verdict is not a disproof of the conclusion but a missing bridge: proof defective; statement unresolved.
• S13. The first clause is S8 at $k=3$. The second: from a model of $\omega_1^2\not\to(\omega_1\omega,3,3,3)^2$, S8 gives a model in which (Q2) fails at $F_3$, hence that ZFC does not prove (Q2) at $F_3$. It does not give a ZFC disproof, which "refutes" suggests. Filed as F2; the intended consistency reading survives.

Premises

• Lemma 2.1 reconstruction (lemma_2_1_reconstruction.md, premise at its reviewed standing, read whole). Interface: for an ordinal $\alpha$, $\alpha\to(\alpha,3)^2$ implies $\alpha\to(\alpha,3,\ldots,3)^2_{k+1}$ for every finite $k\ge1$; its fact 1 (transport along an order isomorphism) and fact 2 (relabeling). The section uses it only through the Tools section and to say it is not needed for S3; its convention covers ordinal targets only.
• Theorem 3.1 reconstruction (theorem_3_1_reconstruction.md, premise, read whole). Interface used: Steps 2 and 3, the restriction of a coloring of $[\omega_1^2]^2$ to $[\omega_1\omega]^2$ and the reading of the result in $\omega_1^2$, with its general remark that a subset of the right type suffices; Theorem A (the imported theorem at $n=3$); Theorem C (relative consistency of $\mathrm{MA}_{\aleph_1}$); and its fidelity note that CH gives $\omega_1\omega\not\to(\omega_1\omega,3)^2$, used as the witness in F1.
• Tools (T1)--(T4) (connections.md, lines 38--93; proofs owned by another reviewer). Interfaces confirmed; (T4) re-derived in W3 with graph targets. (T1)'s phrase "all of whose pairs have color $i$" was read as "edges" for graph targets (F6).
• Baumgartner (1989), main theorem. Not held. Interface: $\mathrm{MA}_{\aleph_1}$ implies $\omega_1\omega\to(\omega_1\omega,n)^2$ for every finite $n$, two colors, the large target in color $0$. Read through the card and result page (which cite the zbMATH review, not read here) and the introduction of Chen, Garti and Weinert, p. 5, page image. The card and the section call the restatement refereed; the held file is arXiv:1801.00238v1, and its own card records an arXiv preprint (F3).
• Baumgartner and Hajnal (1987), negative relation. Not held. Interface: CH gives $\omega_1^2\not\to(\omega_1\omega,4)^2$. Result page read; Komjáth pp. 425 and 442, page images.
• Baumgartner and Hajnal (1987), positive relation. Not held. Interface: $\omega_1^2\to(\omega_1\omega,3,3)^2$ in ZFC. Result page read; Komjáth pp. 425 and 442, page images. Used only for the witness in F1.
• Erdős and Hajnal, CH negative relation. Not held; $\omega_1\omega\not\to(\omega_1\omega,3)^2$ under CH, as reported on the theorem page, the Baumgartner card and its result page from the review. Used only for the witness in F1.
• Folkman (1970), Theorem 1. Held; $f(k_1,k_2)=\max(k_1,k_2)$ for $k_1,k_2\ge2$, with the definitions of p. 19 (finite graphs; $\Gamma$; the dimension $\delta$). Printed p. 20 page image and p. 19 text layer read; proof not read.
• Nešetřil and Rödl (1976). Not held. Interface as recorded on Problem 924 and by Komjáth p. 442 (page image): for a finite $Y$ with no $K_{k+1}$ and $r<\omega$ there is a finite $X$ omitting $K_{k+1}$ such that every $r$-coloring of its edges has a monochromatic induced copy of $Y$. Second-hand at both places.
• Solovay and Tennenbaum (1971). Not held; Theorem C on the theorem page.
• Problem statements of E0597, E1171 and E0924 at this commit.
• Catalog page for Problem 597. Not read (web excluded); the section's attributions to it (the Erdős--Hajnal relation, the original $K_4$-free formulation, Baumgartner's $K_{\aleph_0,\aleph_0}$ relation) are unverified here; Komjáth p. 425 supports the first.
• Explicit assumptions. $\mathrm{MA}_{\aleph_1}$ for S2--S4; the imported theorem at full strength for S2--S4; subgraph copies throughout.

Findings

• F1 (required). Location: section lines 394--399, "(ii) Under $\mathrm{MA}_{\aleph_1}$, whether the proof of $\omega_1\omega\to(\omega_1\omega,n)^2$ extends to countable, then $\aleph_1$-sized, $K_4$-free and $K_{\aleph_0,\aleph_0}$-free $G$: ... a negative outcome, a counterexample $G$ under $\mathrm{MA}_{\aleph_1}$, shows that (Q1) has no positive answer in ZFC and names a third necessary exclusion"; lead lines 135--140, "determine whether the proof for finite $n$ extends ... A negative outcome, a counterexample $G$ under $\mathrm{MA}_{\aleph_1}$, shows that (Q1) has no positive answer in ZFC and names a third necessary exclusion." Defect: the investigation is about the relation on $\omega_1\omega$, so its negative outcome is $\omega_1\omega\not\to(\omega_1\omega,G)^2$ in a model of $\mathrm{MA}_{\aleph_1}$; the conclusion needs $\omega_1^2\not\to(\omega_1\omega,G)^2$ in some model, and a negative relation does not transfer from the initial segment to $\omega_1^2$. Witness: under CH, $\omega_1\omega\not\to(\omega_1\omega,3)^2$ while $\omega_1^2\to(\omega_1\omega,3)^2$ holds in ZFC, so the same pattern of inference at $G=K_3$ from CH would conclude falsely that Problem 597 at $K_3$ has no positive ZFC answer. Replacement: "a negative outcome shows only that this route does not cover $G$; (Q1) has no positive answer in ZFC, and $G$ is a third necessary exclusion, if some model satisfies $\omega_1^2\not\to(\omega_1\omega,G)^2$ for an admissible $G$, which a failure on $\omega_1\omega$ does not give."
• F2 (suggested). Location: section lines 400--402, "a consistent negative answer for that case refutes (Q2) at $F_3$"; lead lines 141--143, "a consistent negative answer to that case refutes (Q2) at $F_3$." Defect: the consequence proved (S8) is that (Q2) fails at $F_3$ in every model of $\omega_1^2\not\to(\omega_1\omega,3,3,3)^2$, hence is not a theorem of ZFC; "refutes" reads as a ZFC disproof. Replacement: "shows that (Q2) at $F_3$ is not provable in ZFC."
• F3 (suggested). Location: section lines 336--338, "the statement is taken from the zbMATH review and one refereed restatement"; lead lines 106--107, "rests on the review and one refereed restatement." Defect: the held restatement is the file arXiv:1801.00238v1 of Chen, Garti and Weinert, dated 31 December 2017, which its library card records as an arXiv preprint; nothing held establishes that it was refereed, and the page's own scope rule cites sources at the standing their library pages record. The Baumgartner (1989) card is the origin of the word. Witness: the first page of the held PDF and the Chen, Garti and Weinert card. Replacement: "one held restatement (an arXiv preprint)", or record the publication data on that card first and cite it at that standing.
• F4 (suggested). Location: section lines 379--380, "The converse implication gives nothing: the relation of Problem 1171 yields (Q2) only for subgraphs of a triangle." Defect: read literally, "only for" claims that no other $G$ is reached, an independence statement with no argument; what is shown is that target weakening reaches exactly the subgraphs of a triangle. Replacement: "In the converse direction, target weakening gives (Q2) exactly for the subgraphs of a triangle; whether the relation of Problem 1171, or ZFC alone, gives (Q2) for any other $G$ is not addressed here."
• F5 (note). Location: lead lines 110--112, "the convention of the lemma page and of Komjáth's survey." Defect: the lemma reconstruction page defines the relation for ordinal targets only; the subgraph convention for graph targets is (T1) in the Tools section of the connections page. Komjáth's usage ($K_4\not\le X$ for subgraph containment, and the reformulation of the CH relation as a $K_4$-free graph on $\omega_1^2$ without an independent set of type $\omega_1\omega$) is consistent with it. Replacement: cite (T1) of the connections page in place of the lemma page.
• F6 (note). Location: Tools (T1), lines 42--43, "a copy of $H_i$ all of whose pairs have color $i$"; section lines 339--341, "$G$ embeds in $K_n$, so $\omega_1\omega\to(\omega_1\omega,G)^2$ by (T2)". Defect: for a graph target the pairs meant are the edges of the copy, as S8 and the lead's definition use it, and the (T2) step replaces the ordinal target $n$ by a graph, which (T2) states only for a target embedding in a target of the same kind. Both are harmless once the identification of an $n$-element homogeneous set with a copy of $K_n$ is written. Replacement: say "all of whose edges have color $i$" in (T1), and in the section "an $n$-element set homogeneous in color $1$ is a copy of $K_n$ in color $1$, which contains a copy of $G$".
• F7 (note). Location: section lines 384--385, "nothing held decides the infinite case." Not a defect: the sentence is true of the corpus. It should not be read as "nothing is known". The reviewer records, from memory and outside the held corpus, that the classical theorem of Erdős and Hajnal that a graph of uncountable chromatic number contains $K_{n,\aleph_1}$ for every finite $n$ would give ZFC instances of (Q1) at $G=K_{n,\aleph_1}$ (bipartite, hence $K_4$-free, and $K_{\aleph_0,\aleph_0}$-free): a graph on $\omega_1\omega$ that is a countable union of independent sets has an independent set of type $\omega_1\omega$, because each block of type $\omega_1$ meets some class in an uncountable set and one class serves infinitely many blocks, so a graph with no such independent set has uncountable chromatic number. This is offered for investigation (ii), is not part of the verdict, and needs the source before use.

Verdict

• S1: not a claim (conventions).
• S2: refutation-failed, conditional on $\mathrm{MA}_{\aleph_1}$ and the imported theorem at full strength.
• S3: refutation-failed, under the same conditions; the consistency step rests on Theorem C.
• S4: refutation-failed, under the same conditions.
• S5: refutation-failed, under the hypothesis it names.
• S6: refutation-failed.
• S7: refutation-failed, resting on Folkman's Theorem 1 (held, statement checked on the page image) and the Nešetřil--Rödl theorem (second-hand).
• S8: refutation-failed.
• S9: refutation-failed under the route reading; the literal reading is not established (F4).
• S10: refutation-failed, with the imported negative relation's hypothesis left open as the section leaves it.
• S11: refutation-failed as a statement about the held corpus.
• S12(a): refutation-failed. S12(b): proof defective; statement unresolved (F1).
• S13: refutation-failed for the consistency reading; the word "refutes" overstates it (F2).
• S14: not a claim; F3 and F5 record the fidelity corrections, and the lead's reference to the first section's CH obstruction was not assessed.

Limitations: the chapter of Baumgartner (1989) and the paper of Nešetřil and Rödl are not held, and the catalog page for Problem 597 was not consulted, so the three catalog attributions in the section's first paragraph and the full-strength statement of the imported theorem are taken at the standing their library pages record. The first section of the connections page, which the lead cites for a CH obstruction, is outside this review's read set. This focused review assigns no tier and changes no status.