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Subject

The pages as they stood at 2026-09-28T05:03:27Z: wiki/research/erdos_1219/lemma_1_1_reconstruction.md, wiki/research/erdos_1219/theorem_1_2_reconstruction.md and wiki/research/erdos_1219/corollary_1_3_reconstruction.md, each read whole as of that time. Reports: the Lemma 1.1 review, the Theorem 1.2 review and the Corollary 1.3 review, each read whole.

Read for adjudication, as of the same time unless stated: the sections "Independence and the assignment", "Exact subjects and durable evidence", "Report contract", "Grading and claim standing", "Whole-claim report" and "Audit checklist" of docs/verification.md, and the section "Source fidelity" of docs/evidence.md; the held scan of Shelah (1975) (twenty pages, 3,493,641 bytes, matching the card's provenance line, without a text layer), PDF pp. 1--5 (printed pp. 1257--1261) rendered at 200 dots per inch and read in full, with crops at 400 dots per inch read for clause (3) and the Remark of Lemma 1.1 (p. 1258), the two printed type counts and the printed application of the hypothesis (p. 1259), and the application of the relations from [4], the sentence invoking Lemma 1.1 and the Remark after Corollary 1.3 (p. 1260), and PDF p. 19 (printed p. 1275) rendered at 150 dots per inch and read for the entries [1], [4] and [7] of the reference list; the held PDF of Komjáth (2025) at printed pp. 419 and 442 (PDF pp. 2 and 25), from the text layer and rendered at 110 dots per inch and read; the Shelah card and its result pages theorem_1_2 and corollary_1_3; the Komjáth card; the whole of Problem 1219; the folder index; and the grade record of a sibling folder, for the shape of this record only. No other review, workspace file or web page was read.

Independence, by role: distinct grader in a fresh context, given only this assignment. The grader wrote none of the three pages, no page of the folder or of the library cards named above, and none of the three reports, and had no communication with the author or with any of the three reviewers. A grader is not blind: the standing text of the cards, the folder index, the problem page and all three reports were read by design.

Exposure ruling. Each report discloses that its reads returned more than its allowed sections. The Lemma 1.1 review read the Shelah card whole (its read-status paragraph, a Contents bullet summarizing the structure of the lemma's proof and quoting the printed type count, its Compiled scope and a Bears-on sentence on the problem's status) and the result page theorem_1_2 whole (Proof pointer, Dependencies and Bears-on). The Theorem 1.2 review read the Shelah card's read-status paragraph and the Komjáth card's citation and digest paragraphs. The Corollary 1.3 review saw the problem page's status field, the Theorem 1.2 page's Standing paragraph, the result page's Read-depth paragraph and the Shelah card's read-status sentence. Content test: nothing in any report could only have come from that text. Every finding carries a witness on a page image or on the frozen page: the Lemma 1.1 review's F4 and F6 rest on crops of p. 1259 at 400 dots per inch, its F1 and F2 on p. 1258, and its clause (3) attack on the page's own Standing paragraph; the Theorem 1.2 review's F1 and F4 rest on the images of printed pp. 1260 and 442; the Corollary 1.3 review's F2 rests on the Imported results section of the Theorem 1.2 page, which was in its allowed set, and its strongest attack on the print of the corollary. The direction of every attack follows from the subject page and the print, and none of the exposed text is a review of, or a verdict on, any of the three pages. The exposures are ruled immaterial for all three reports.

Reports graded

Lemma 1.1 review: pass. The subject block resolves (subject date and path, the path unchanged; the artifact identified with its page mapping and the rendering resolutions). The independence facts and two exposures are stated. The restatement carries the convention, all the data, the growth conditions, the hypothesis (H) with its four quantifiers, and the four clauses of the conclusion with their quantifiers, and it separates the reading under which clause (3) is proved. The checklist is filed against the shared page's canonical modes and named patterns rather than under the ten item names of the Erdos checklist; each of the ten items nevertheless carries an explicit verdict: quantifiers and scope under the "almost all" and "exceptional sets" modes, with the boundary cases in the strongest attack; circularity under the circular-use and termination modes; model and convention changes under the relaxed-system and model-class modes; finite and statistical overreach under the heuristic and finite-verification modes; uniformity, extremal conclusions and consequences and composition under their named patterns; computation under the certified-bracket pattern; reproduction under the reproducibility, decoration-leg and gate patterns; and source and verdict fidelity under the verifier-quotation pattern together with the explicit "Source fidelity" verdict of its Verdict section. The three weakest steps are re-derived, not paraphrased: the exceptional set with the two counts and the regularity use, the bridge through $a^*_\beta$ with the exact patterns and the two type equalities, and clause (3) with the fiber argument and its composition after the recursion; the grader re-derived each from the page images and agrees. The strongest attack is real: an instance in which $P_\alpha$ forces $|B_\alpha|=\aleph_0$ while $2^{\chi+\kappa}<\operatorname{cf}\mu(\alpha)$, under which the fiber step fails and the page's proof of the printed clause (3) does not go through; the report correctly places the outcome as a labeling finding, because the page states its reading and the clause is unused downstream. Three further attacks (the printed chain at $\alpha=0$, a wrong version of the bridge, the degenerate parameters) are recorded with their outcomes. The premises carry their interfaces and reading depth: no consumed local claim, six cardinal-arithmetic facts each checked, the source interface with locators, and the consumer page read for one sentence only. The verdict is stated in full and assigns no tier.

Theorem 1.2 review: pass. The subject block resolves (subject date and path, both artifacts with page mappings and resolutions). The independence facts and two exposures are stated. The restatement carries the three hypotheses with explicit quantifiers, the reading of the printed bound, the sum with its index set, both conclusions and the convention, and it states the scope facts it confirmed. All ten checklist items carry an explicit verdict by name, with the inapplicable ones marked. The three weakest steps are re-derived: the arithmetic hypotheses of Lemma 1.1, including $\lambda_i^{\mu(i)}=\lambda_i$ and the growth product; Step 4 with the use of (ER), (N) and the corrected bound; and $|B|=\lambda$ from the unboundedness of $I$; the grader re-derived each and agrees. The strongest attack is real and placed where the risk sits: the application of Lemma 1.1 from the side of (H) and from the side of the arithmetic hypotheses, a counterexample ($2^{\aleph_n}=\aleph_{n+1}$, $\lambda=\aleph_\omega$) showing that the printed bound cannot be what the proof proves, and an attack on the imported relation that ended in the reviewer's own derivation of it. That derivation, by an elementary submodel of size $2^\mu$ closed under $\mu$-sequences and a recursion of length $\mu^+$, was checked here step by step and is correct. The premises record Lemma 1.1 through its statement, (ER), (K), (S) and (EDM) with interface, source and reading depth, the cardinal-arithmetic facts, and the reading of the printed bound. One statement in its F4 and in its Premises is wrong and is rejected below; it does not touch the verdict. The verdict is stated in full and assigns no tier.

Corollary 1.3 review: pass. The subject block resolves (subject date and path; both artifacts with page mappings). The independence facts and three exposures are stated. The restatement carries the partition convention, the corollary with its chain and the reading of the sequence, the supplied consequences, and the page's second claim with its quantifier over sequences. All ten checklist items carry an explicit verdict by name. The three weakest steps are re-derived: the three hypotheses of Theorem 1.2 at $\lambda=\aleph_\omega$ with the explicit threshold and the choice of $m$ and $k$, the cardinal $\chi$ with the sum formula itself re-derived, and the identification of the two sums by two inequalities; the grader re-derived each and agrees. The strongest attack is real: the range of $n(k)$, with an instance in which $n(0)\ge\omega$ would make the chain say nothing about the powers below $\aleph_\omega$ and the conclusion fail, and the printed bound of Theorem 1.2. The premises record Theorem 1.2 at its interface with its standing as read, Ramsey's and Sierpiński's theorems, the sum formula, Komjáth's Problem 3 and the catalog statement, each with reading depth. The verdict is stated in full and assigns no tier.

Corrections

C1. Page: lemma_1_1_reconstruction.md. Location: frontmatter title and desc. Replace the title with "Canonization Lemma 1.1: canonizing functions on fast-growing blocks", and replace the last clause of the desc, "so that a value depends only on the blocks of its arguments.", with "so that a value with one argument from each of the two highest blocks used and the rest from lower blocks does not depend on which elements of those two blocks are taken; a two-place function then depends only on its two block indices." Basis, checked against p. 1258: clause (1B) reads $F_i(b,c,a_1,\ldots)=F_i(b',c',a_1,\ldots)$ with $a_1,\ldots$ fixed elements of $\bigcup_{i<\alpha}B_i$; the value of a function of three or more places may still change with those parameters, and only clause (3), under its extra hypotheses, removes that dependence. The desc and the title claim the conclusion of (3) for the whole lemma, and the desc propagates into the generated index row. The Lemma 1.1 review filed this as F3 at severity suggested; it is accepted as a correction on the grader's own verification, because a summary of a result is a claim surface and this one is stronger than the result. The change touches the frontmatter only.

C2. Page: theorem_1_2_reconstruction.md. Location: section "Reading notes", after the second bullet. Add the bullet: "The printed proof places the two homogeneous sets in $A_i$ (p. 1260: 'there are sets $B_{i,0},B_{i,1}\subseteq A_i$ of cardinality $\mu(i)$') in the sentence that applies the two relations from [4] to every $A'_i\subseteq A_i$ of size $\lambda_i$; they are read as subsets of that $A'_i$, the form that the lemma's hypothesis (H) needs and that the Step 4 claim states and proves." Basis, checked on the crop of p. 1260 at 400 dots per inch: the printed sentence reads exactly as quoted, and (H) requires the sets inside the given $C$. The page's Step 4 proves the needed form; its Reading notes record the neighboring slip $|B_\alpha|=\mu(i)$ but not this one, while counting Step 4 among the steps that follow the printed proof. The Theorem 1.2 review filed this as F1 at severity required; accepted. The reviewer's second part, a clause added to the last paragraph of Step 5, is optional: the bullet records the reading, and the Step 5 sentence concerns the arithmetic items.

C3. Page: theorem_1_2_reconstruction.md. Location: section "Preliminary: $\kappa<\lambda$", the sentence "$\mu$ is infinite, since $2^\mu$ is not." Replace it with "$\mu$ is infinite, because $2^\mu\ge\lambda$ is infinite while $2^\mu$ is finite for finite $\mu$." Basis: the sentence as written gives as its reason that $2^\mu$ is not infinite, which is false, since $2^\mu\ge\lambda$; the intended reason is the one the replacement states, and the conclusion is right. The paragraph is supplied by the page, so no source is involved. The Theorem 1.2 review filed this as F2 at severity suggested; accepted on the grader's own verification, because the sentence is false as written.

C4. Page: theorem_1_2_reconstruction.md. Location: section "Imported results", bullet (ER), from "is quoted as the Erdős--Rado theorem in Komjáth's survey, printed p. 442, PDF p. 25, in the commentary on Problem 53," through "; that held page anchors the statement, not its proof." Replace those words with: "is printed in Komjáth's survey, printed p. 442, PDF p. 25, in the commentary on Problem 53, komjath_2025_erdos_hajnal_problem_list, inside a remark the survey attributes to Erdős and Hajnal; the survey's next paragraph attaches the label Erdős--Rado to the form $\lambda^+\to(\lambda^+,(\kappa^+)_\kappa)^2$ for cardinals $\lambda$ with $\lambda^\kappa<\lambda^{\kappa^+}$. That held page anchors the statement, not its proof." Basis, checked on the text layer and the image of printed p. 442: the relation $(2^\kappa)^+\to((2^\kappa)^+,(\kappa^+)_\kappa)^2$ appears in the sentence "Erdős and Hajnal remarked that if $\kappa$ is infinite, then ...", and the label "(Erdős--Rado)" is attached in the following paragraph to $\lambda^+\to(\lambda^+,(\kappa^+)_\kappa)^2$ after "let $\lambda$ be such that $\lambda^\kappa<\lambda^{\kappa^+}$". The page's sentence characterizes the survey as labeling the relation it quotes, and the survey does not. The Theorem 1.2 review filed this as F4 at severity note, with a replacement that is not adopted (see below); the defect is accepted as a correction on the grader's own verification, because the characterization of a source is a checklist item.

Rejected and downgraded findings

• Lemma 1.1 review, F1 (suggested: say at the Statement that clause (3) is proved under the reading $|B_\alpha|=\mu(\alpha)$). Downgraded to optional; no change required. The Statement transcribes the printed clause, which is what fidelity asks, and the page states the reading twice, in its Standing paragraph at the head of the page and at the head of the clause (3) section, with the reason ("The cofinality hypothesis has no force unless $|B_\alpha|=\mu(\alpha)$"). The grader confirms the mathematics: with $|B_\alpha|=\nu$ and $\operatorname{cf}\nu\le2^{\chi+\kappa}$ every fiber of $H_\alpha$ may be small, so the reading is load-bearing for the page's proof of (3), and whether the printed clause holds without it by another argument is a question about the source that neither the report nor this grade settles. The proposed sentence is accurate and may be added.
• Lemma 1.1 review, F2 (suggested: add the Remark's second sentence). Downgraded to optional; no change required. The print (p. 1258) does carry "We can assume that the range of $F_i$ is $2^\chi$", and the reviewer's justification through the count (A) is right, since $(2^\chi)^\theta=2^\theta$. The page's paraphrase of the Remark is incomplete, not wrong, and the sentence is used neither by the proof nor by the application.
• Lemma 1.1 review, F4 (suggested: record that the printed first count "$\le2^{|B|+\chi}$" lacks $\aleph_0$). Downgraded to optional; no change required. The witness is right: on p. 1259 the count is printed without $\aleph_0$, and with $\chi=3$, three one-place functions into $3$ and $B=\emptyset$ there are $27$ types against $2^{|B|+\chi}=8$. The page does not transcribe that count; its bound (A) is supplied with its own proof, the printed count is correct whenever $\chi$ is infinite, and under the lemma's hypotheses the difference is absorbed by $\kappa\ge\aleph_0$ in every bound the proof uses. The proposed note is accurate and may be added beside the two printed slips the page records.
• Lemma 1.1 review, F5 (note: the chain in (A) fails at $\chi=0$). Downgraded to optional; no change required. At $\chi=0$ there are no functions, exactly one type, and $2^{\chi\cdot\theta}=1\ne2^\theta$, so the middle equality of the chain does fail there; the bound $2^\theta$ that (A) asserts holds in every case, and the lemma has no content at $\chi=0$.
• Lemma 1.1 review, F6 (note: the printed "$\alpha<i<a$" on p. 1259 is unrecorded). Downgraded to optional; no change required. The crop confirms the misprint. The page transcribes the bound correctly as $\kappa$, and a typographical slip in an index bound changes no statement; the note may be added for consistency with the slips the page records.
• Lemma 1.1 review, F7 (note: the length phrase in (1) can be read as one length for both clauses). Rejected; no change. The sentence fixes the length as the one "that fills the remaining places of $F_i$", which is $n_i-1$ under the display (1A) and $n_i-2$ under (1B), and the displays follow immediately.
• Theorem 1.2 review, F3 (suggested: the Sierpiński citation may cover only the countable case). Downgraded to an open citation question; no change required on held evidence. The 1933 note is not held and a web search is outside this grade's read set, so neither the report nor the grade can check its scope; the relation $2^\mu\not\to(\mu^+)^2_2$ for every infinite $\mu$ is standard, its use in the supplied preliminary is correct, and the reviewer's proposed qualification is safe to add.
• Theorem 1.2 review, F4, the proposed replacement text. Rejected as worded, with the defect accepted as C4. The replacement says that $\lambda=2^\kappa$ is an instance of the survey's condition $\lambda^\kappa<\lambda^{\kappa^+}$, and the report's Premises repeat this. For $\lambda=2^\kappa$ the condition reads $2^\kappa<2^{\kappa^+}$, which is not a theorem of ZFC: $2^{\aleph_0}=2^{\aleph_1}$ is consistent. C4 therefore records what the survey prints without asserting that instance.
• Theorem 1.2 review, F5 (note: two listed standard facts are unused). Downgraded to optional; no change required. Checked: neither $2^{\sum_i\kappa_i}=\prod_i2^{\kappa_i}$ nor the bound on a union of fewer than $\operatorname{cf}\theta$ small sets is used on the theorem page; both are used on the lemma page. The list may drop them.
• Theorem 1.2 review, F6 (note: the cofinal sequence of cardinals also needs $\lambda$ to be a limit cardinal). Downgraded to optional; no change required. The observation is right, and the condition holds where the sentence stands, since the preliminary has already shown $\lambda$ singular; the proposed phrase may be added.
• Corollary 1.3 review, F1 (suggested: mark the normalization of "Theorem 2"). Downgraded to optional; no change required. The print reads "Theorem 2" (p. 1260) and the paper has no result of that number; the Theorem 1.2 page of the folder and the library card both record the misnumbering, and the corollary page's paraphrase is correct.
• Corollary 1.3 review, F2 (suggested: list Sierpiński's theorem and the inherited imports in the Standing). Downgraded to optional; no change required. The Standing describes this page's own reconstruction, the specialization and the identification of the sums, for which Ramsey's theorem is the only import; Sierpiński's theorem enters only a remark on the finite-sequence boundary case, attributed to the problem page, and the imports of the Theorem 1.2 page are listed on that page and in the folder index. The proposed sentence is accurate and may be added.
• Corollary 1.3 review, F3 (note: the reading $n(k)<\omega$ is unmarked). Downgraded to optional; no change required. The print gives no range, and the reading is the ordinary meaning of the notation, forced by the summation index $n<\omega$, matched by Komjáth's "$(n_i<\omega)$" (p. 419) and noted on the corollary's result page; the reviewer's instance with $n(0)\ge\omega$ is a correct account of why the reading is load-bearing, and the proposed bullet may be added.

Checks of the grader's own that produced no correction. The locators on all three pages hold: printed p. $n$ is PDF p. $n-1256$ of the twenty-page scan; Lemma 1.1 with its Remark is on p. 1258 and its proof runs to the top of p. 1260; Theorem 1.2, its proof, Corollary 1.3 and the Remark are on p. 1260; Conjecture 1A is on p. 1261; the § 0 statement is on p. 1257; the reference list on p. 1275 gives [1] as the Erdős--Hajnal problem list, [4] as Erdős, Hajnal and Rado, Partition relations for cardinals, Acta Math. Acad. Sci. Hungar. 16 (1965), and [7] as Shelah's Notes in combinatorial set theory; Komjáth's Problem 3 is on printed p. 419, PDF p. 2, with "$(n_i<\omega)$" and the sum $2^{\aleph_{n_0}}+2^{\aleph_{n_1}}+\cdots$. The quotations the pages carry from the print (the two relations cited to [4]; "$B_\alpha\subseteq A$, $|B_\alpha|=\mu(i)$"; "Theorem 2"; the chain "$\le2^\chi\cdot\prod_{i<\alpha}2^{\mu(i)}\le\lambda^\alpha$"; the second-stage count) are verbatim. The reading "eventually $\ge\lambda$" is forced: with $2^{\aleph_n}=\aleph_{n+1}$ for all $n$ the printed form would assert $\aleph_\omega\to(\aleph_\omega)^2_2$, which fails for every singular cardinal. The lemma page's type equivalence, defined through patterns with every placement of the variable, coincides with the source's $\operatorname{tf}$ on single elements once the family of functions is closed under permutations and identifications of variables, as the source assumes without loss of generality.

Graded verdicts

• lemma_1_1_reconstruction.md: fidelity faithful, with the summary correction C1 in the frontmatter; argument sound for clauses (1A), (1B) and (2), and for clause (3) under the reading $|B_\alpha|=\mu(\alpha)$ that the page states, which is unused downstream. The counts (A)--(C), the exceptional set, the two thinnings and the three verifications were re-derived here from the page images.
• theorem_1_2_reconstruction.md: fidelity faithful, with the reading correction C2, the wording correction C3 and the citation correction C4, none of which touches the statement; argument sound, with the Canonization Lemma consumed at the interface the lemma page states and with (ER), (S) and (EDM) as identified external premises that are not held. The preliminary, the choice of the $\mu(i)$, the block arithmetic, the two uses of (ER), the hypotheses of the lemma, the Ramsey step and the three-color derivation were re-derived here.
• corollary_1_3_reconstruction.md: fidelity faithful; argument sound, its standing bounded by that of the Theorem 1.2 page. The three hypotheses at $\aleph_\omega$, the cardinal $\chi$ and the identification of the two sums were re-derived here.

No tier is assigned and no status changes.