---

Subject and independence

Role: independent reviewer in a fresh context, given only the commissioning assignment. The reviewer took no part in writing the page or any page in its folder and had no communication with the page's author. Charge: refutation.

Frozen subject: path wiki/research/erdos_1171/lemma_2_1_reconstruction.md as it stood on 2026-09-28T05:03:27Z (called "the commit" below), read at that commit. The path read is the one named.

Artifact: the held PDF under the library card Gao (2026) (gao_2026_finite_color_partition_relation_omega_1_squared.pdf, four pages, 244,055 bytes, matching the card's provenance line). Physical pp. 1--4 were read in full from the text layer; pp. 2--3 (Lemma 2.1 and its proof) were followed line by line, and p. 1 (the definition of the arrow relation) and p. 4 (Remark 3.2) were read at the depth of the cited sentences. All four pages were rendered to images at 130 dpi and read, so every displayed formula (the abstract's relations, the definition on p. 1, the statement of Lemma 2.1, the colorings $c$ and $c'$ and the two-coloring $d$ on pp. 2--3, Theorem 3.1 and Remark 3.2) was checked against the image and not only the extraction. No canonical conversion sits beside the PDF.

Allowed material actually read: the page; the library card's provenance paragraph; the Statement section of the result page lemma_2_1 under the card; the region of wiki/problems/set_theory/E1171/_index.md above its "Current assessment" heading (the page has no Statement heading, so the problem statement was read as the part preceding that heading); and docs/verification.md ("Report contract", "Audit checklist", "Whole-claim report" and the Erdos-specific "Audit checklist"), docs/evidence.md ("Source fidelity") and docs/math_authoring.md, all at the frozen commit. The page names no reconstruction page as an input ("Depends on. Nothing beyond the hypothesis"), so no sibling reconstruction was read; the two Baumgartner library cards were not needed and were not read.

Exposures: three, disclosed here. (1) The library card was printed whole, not only its provenance paragraph; its "Claim type", "Fidelity to Problem 1171" and "Read status" paragraphs and its standing sentence ("unrefereed, proofs followed, no independent review") reached the reviewer. (2) The result page lemma_2_1 was printed whole; its Source paragraph's standing sentence and its "Rewritten proof" and "Check performed here" sections reached the reviewer. (3) The problem statement region of E1171.md carries a one-line status field ("Not disprovable") and a Source paragraph describing the site's label and proof-claims tab. None of these is another review; the mathematics below was re-derived from the PDF alone, and the exposed text changed no verdict. The folder _index.md, every evidence/ folder, the sibling reconstruction, all assessment and standing text beyond the exposures listed, everything among the private working files, other reviews and web searches were not read.

Restatement

Convention (source p. 1; page "Definitions"). For an ordinal $\alpha$, ordinals $\beta_0,\ldots,\beta_{n-1}$ and an integer $n\ge1$, the relation $\alpha\to(\beta_0,\ldots,\beta_{n-1})^2_n$ holds when every function $c:[\alpha]^2\to\{0,\ldots,n-1\}$ (a coloring with $n$ colors, not required to take every value) admits an index $i<n$ and a set $X\subseteq\alpha$ of order type $\beta_i$ on which $c$ is constantly $i$. The subscript counts the colors and is dropped for $n=2$; a set of ordinals is ordered by membership, and a set of order type $3$ is a three-element set.

Proposition (Lemma 2.1, p. 2). For every ordinal $\alpha$, finite or infinite, with no restriction: if $\alpha\to(\alpha,3)^2$, then for every integer $k\ge1$ the relation

$$\alpha\to(\alpha,\underbrace{3,\ldots,3}_{k})^2_{k+1}$$

holds, that is, every $c:[\alpha]^2\to\{0,\ldots,k\}$ has a set of order type $\alpha$ on which $c$ is constantly $0$ or a three-element set on which $c$ is constantly some $i\in\{1,\ldots,k\}$. The hypothesis is an explicit premise of a conditional statement; the lemma asserts nothing about which $\alpha$ satisfy it. Nothing beyond the hypothesis is used, and no axiom beyond ZF enters: the argument uses induction on $k$, order isomorphisms between sets of ordinals and their order types, and function definitions.

Checklist

• Quantifiers and scope: pass. "For every finite $k\ge1$" is proved by induction with each $P(k)$ a statement about all colorings; $\alpha$ is an arbitrary ordinal in the source and on the page; the boundary $k=1$ is the hypothesis; no "almost all" appears.
• Circularity: pass. The step from $P(k)$ to $P(k+1)$ uses $P(k)$ on the merged coloring and the fixed hypothesis on a two-colored set; neither is the target $P(k+1)$.
• Model and convention changes: pass. The page's convention equals the source's (p. 1) clause for clause; the only transformation of objects is the transport of a coloring along an order isomorphism, which the page proves (fact 1).
• Finite and statistical overreach: inapplicable; no finite case or heuristic average is used as a proof.
• Uniformity: pass. There are no constants or error terms; the one parameter dependence, that the triangle colors of $P(k+1)$ are exactly $\{1,\ldots,k+1\}$, was recomputed below.
• Extremal conclusions: inapplicable; the lemma asserts no infimum, supremum or sharpness.
• Consequences and composition: pass. The "Consumed by" line names Theorem 3.1 with $\alpha=\omega_1\omega$, which matches the source's use (p. 3); the lemma exports exactly its conditional statement, and the page adds no "hence" beyond the two Checks-and-scope explanations re-derived below.
• Computation: inapplicable; no computation is involved.
• Reproduction: inapplicable; no rerun command or coverage claim is made.
• Source and verdict fidelity: pass with one suggested correction (F1). The statement, the locators (physical p. 2 for the statement, pp. 2--3 for the proof, section 2 "A general color-reduction lemma", Remark 3.2 on p. 4, printed page numbers equal to physical), the quoted convention and the sentence that the source calls the lemma standard all match the PDF; the sentence that the source "records" the inessentiality of the target $3$ in Remark 3.2 attributes more to that remark than it says.

Weakest steps

1. Transport of the hypothesis to $Y$ (fact 1, used in alternative 2). The source writes only "Since $\operatorname{otp}(Y)=\alpha$ and $\alpha\to(\alpha,3)^2$, there is either ..." (p. 3). Re-derivation: $Y$ is a set of ordinals, so it is well ordered by membership and there is an order isomorphism $\pi:\alpha\to Y$. Given $d:[Y]^2\to\{0,1\}$, the function $d'(\{\xi,\eta\})=d(\{\pi\xi,\pi\eta\})$ is defined on all of $[\alpha]^2$ because $\pi$ is injective, so $\{\pi\xi,\pi\eta\}$ has two elements. The hypothesis gives $i<2$ and $X\subseteq\alpha$ of order type $\alpha$ (if $i=0$) or $3$ (if $i=1$) with $d'\equiv i$ on $[X]^2$. Put $H=\pi[X]\subseteq Y$; $\pi\upharpoonright X$ is an order isomorphism onto $H$, so $\operatorname{otp}(H)=\operatorname{otp}(X)$, and every pair of $H$ is $\{\pi\xi,\pi\eta\}$ for a unique pair $\{\xi,\eta\}$ of $X$, so $d\equiv i$ on $[H]^2$. This composes with the surrounding argument because $H\subseteq Y\subseteq\alpha$ and $d=c$ on $[Y]^2\supseteq[H]^2$, so $H$ is homogeneous under $c$ as a subset of $\alpha$.

2. Reading the original color off the merged one (alternative 1). The page's $c'$ sends $\{0,1\}$ to $0$ and $j\in\{2,\ldots,k+1\}$ to $j-1$. Re-derivation: if $c'(p)=i\ge1$ then $c(p)\notin\{0,1\}$, since those values map to $0$; hence $c(p)\in\{2,\ldots,k+1\}$ and $c'(p)=c(p)-1$, so $c(p)=i+1$. A set on which $c'$ is constantly $i\in\{1,\ldots,k\}$ is therefore a set on which $c$ is constantly $i+1\in\{2,\ldots,k+1\}$. The source states the same in the words "colors $j\ge2$ are unchanged from $c$" (p. 3) in its own naming; the two namings correspond under $0'\mapsto0$ and $j\mapsto j-1$, which preserves the target list $(\alpha,3,\ldots,3)$ position by position.

3. The color bookkeeping of $P(k+1)$. $P(k)$ concerns colorings into $\{0,\ldots,k\}$, and $c'$ is one, so $P(k)$ applies and returns either a set of order type $\alpha$ in color $0$ or a triangle in a color of $\{1,\ldots,k\}$. Alternative 1 delivers triangle colors $\{2,\ldots,k+1\}$ under $c$; alternative 2 delivers color $0$ on a set of order type $\alpha$ or a triangle of color $1$. The union $\{1\}\cup\{2,\ldots,k+1\}=\{1,\ldots,k+1\}$ is exactly the set of triangle colors of $P(k+1)$, whose colorings take values in $\{0,\ldots,k+1\}$; no color is missed and none lies outside the range.

Strongest attack

The attack was to find a coloring on which the induction step returns an object that does not witness $P(k+1)$: a homogeneous set whose color under $c$ cannot be recovered from its color under $c'$, or a set on which the hypothesis is invoked at an order type other than $\alpha$. The first fails because the merge is injective on $\{2,\ldots,k+1\}$ and only the merged color $0$ has a two-element preimage, and that case is exactly the one the page hands to the hypothesis after restricting $c$ to $[Y]^2$, where $c$ takes only the values $0$ and $1$. The second fails because $P(k)$ is applied with first target $\alpha$, so the set $Y$ it returns has order type exactly $\alpha$, and fact 1 moves the hypothesis onto $Y$ without loss.

Boundary attacks also failed. Finite $\alpha$ and $\alpha\le1$ are covered verbatim: for $\alpha\le1$ every relation with first target $\alpha$ holds with $X=\alpha$, whose pair set is empty, and for finite $\alpha\ge2$ the hypothesis is false (color one pair $1$ and the rest $0$), so the implication is vacuous. "Order type $3$" and "three-element set" coincide for sets of ordinals. A coloring "with $n$ colors" need not be onto, and $P(k)$ quantifies over all functions into $\{0,\ldots,k\}$, so applying it to a $c'$ that omits a value is legitimate. The page's two explanatory claims in "Checks and scope" were attacked as consequence sentences: replacing the target $3$ by any ordinal $\beta$ leaves every line of the induction intact, so "the induction never uses that a triangle has three points" is true; and with a hypothesis $\alpha\to(\beta,3)^2$ for $\beta<\alpha$, the induction hypothesis returns a set of order type $\beta$ to which neither the hypothesis nor its transport applies, so the explanation of why the large target equals the ambient ordinal is correct as a statement about this proof. No defect in the statement, the locators or the argument was found; the one surviving finding (F1) concerns a characterization of Remark 3.2.

Premises

• The hypothesis $\alpha\to(\alpha,3)^2$: an explicit assumption of the conditional statement, not a consumed claim; assumed, not certified.
• Imported theorems: none. The page says so ("no external theorem is imported"), and the re-derivation confirms it; the facts used are the existence of an order isomorphism between a set of ordinals and its order type (ZF) and induction on the integers.
• Fact 2 (relabeling of colors together with the target list): stated on the page without proof, and not load-bearing, since the page's $c'$ is defined directly into $\{0,\ldots,k\}$ and $P(k)$ is applied to it verbatim. Its proof is one line: given a permutation $\sigma$ of the colors, apply the relation to $\sigma\circ c$ and read the index back through $\sigma^{-1}$.
• Consumed local claims: none. The page consumes no L-claim.
• Source held: yes, the PDF named above, at the reading depth stated under Subject and independence. The source is an unrefereed deposit; its standing is neither used nor changed by this review.

Findings

F1.

• Severity: suggested.
• Location: "Checks and scope", bullet "What the number $3$ contributes", the sentence "The source records this only in the form of Remark 3.2."
• Defect: the sentence attributes to the source a record that the value $3$ is inessential. The page's own observation, that the induction never uses that a triangle has three points, is correct, but it is the page's and not the source's.
• Witness: Remark 3.2, physical p. 4, says only that the proof of Lemma 2.1 shows the property $\alpha\to(\alpha,3)^2$ to be "stable under adding finitely many colors with target 3"; it restates the lemma for target $3$ and says nothing about other targets.
• Proposed replacement: "The source does not state this; Remark 3.2 (p. 4) only restates the lemma as the stability of $\alpha\to(\alpha,3)^2$ under adding finitely many colors with target $3$."

F2.

• Severity: note.
• Location: "Definitions", the sentence "Two facts about the relation are used below without further comment" and fact 2.
• Defect: fact 2 is stated without proof and announced as used, but the proof never relies on it; it serves only the comparison with the source's color names ($0'$ and $2,\ldots,k+1$), since the page's $c'$ maps into $\{0,\ldots,k\}$ and $P(k)$ applies to it as stated.
• Witness: the proof's only appeal to it is the sentence "the renaming here is the relabeling of fact 2 and changes nothing."
• Proposed replacement: either add the one-line proof (compose a coloring with the renaming and read the index back through its inverse) or say that fact 2 is used only to relate the page's color names to the source's, so that the proof stands without it.

Verdict

Source fidelity: faithful. The statement, its hypotheses, quantifiers and convention, the locators and the characterization of the proof all match physical pp. 1--4 of the held PDF; the one correction proposed (F1) is a wording change in a commentary bullet, not a change to the statement or the argument.

The argument as reconstructed: sound. Every deduction of the induction was re-derived above; the supplied steps (the transport of fact 1 and the explicit renaming) are labeled as supplied on the page, and nothing the source proves is altered or strengthened.

Limitations: the review covers Lemma 2.1 only and not the source's Theorem 3.1 or the sibling reconstruction; the "folder index" lead cited in the last Checks-and-scope bullet was not checked, since the folder index is outside the read set; the exposures listed under Subject and independence were disclosed and did not affect the verdict.

This focused review assigns no tier and changes no status.