---

Subject

The pages as they stood at 2026-09-28T13:50:48Z. Page: wiki/research/erdos_1219/connections.md, read whole as of that time; its H2 headings stand at lines 39 (Tools), 109 (Problem 1219), 171 (Problem 1220), 286 (Hajnal's Conjecture 1A), 446 (Problem 1218), 511 (Problem 1167), 542 (Other relations) and 569 (Outcomes), which fixes the line ranges the reports name. Lead: wiki/research/leads/triple_canonization_for_hajnal_conjecture/_index.md, read whole as of that time; it is the only lead the page files. Reports, each read whole: the Tools review, the Problem 1219 review, the Problem 1220 review, the Conjecture 1A review, the Problem 1218 review, the Problem 1167 review and the Other relations review.

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 "Source fidelity" of docs/evidence.md; the reconstruction pages Lemma 1.1 and Theorem 1.2, read whole (the Corollary 1.3 page enters no finding and was not re-read; its standing is taken from the earlier grade, read for that and for the shape of this record); the folder, evidence and verify indexes; Problem 1219 whole, in particular its Known Results bullet on the site remark, and the Statement blocks of Problem 1220 (through its Progress section), Problem 1218, Problem 1167, Problem 1168, Problem 70 and Problem 1172; the section "Other relations in the set-theory folder" of the Problem 1171 connections page and the heading list and "Other pages checked" section of the Problem 501 connections page; the held PDF of Komjáth (2025) at printed pp. 419--420 (PDF pp. 2--3) on the text layer, and p. 419 rendered at 300 dots per inch and read at Problems 1 and 2; the held PDF of Erdős and Rado (1956) at printed p. 432 (PDF p. 6) on the text layer. Not read: the Shelah scan (no finding turns on it, and the earlier grade checked the locators the reports use), the catalog pages (web excluded), and the sources [45], [170], [171], Erdős--Hajnal--Rado (1965) and [ErHa71], none held.

Independence, by role. Each report was written by an independent reviewer in a fresh context, given only its commissioned section, the page's Scope and Tools paragraphs as inputs, the reconstruction pages as premises, the named problem statements and library material, and the wiki pages; no reviewer wrote any page of the folder or of the lead, and no reviewer read another review, the earlier grade, or the standing text of the subject. The grader is distinct from the author and from all seven reviewers, works in a fresh context given only this assignment, wrote none of the pages or reports, and had no communication with the author or with any reviewer. A grader is not blind: the standing text of the problem pages, the folder indexes, the earlier grade and all seven reports were read by design.

Exposure rulings. The Tools review read the Komjáth and Shelah cards past their provenance paragraphs, a read-depth sentence on the result page theorem_1_2 and the Status lines and one References list of four problem pages. The Problem 1219 review read the two cards and the result page in full and saw the file names under evidence/verify/. The Problem 1220 review had the Current assessment and Progress sections of Problem 1220 displayed (author-recorded notes that paraphrase the section under review), saw the neighboring Problem 1219 section, and read the result page past its Statement. The Conjecture 1A review reports no exposure beyond file listings. The Problem 1218 review read the survey card whole, the Source and Standing paragraphs of the theorem page, heading listings and a status listing of file names. The Problem 1167 review saw the Standing, Proof pointer and Bears-on parts of the result page conjecture_1a (three sentences, one naming a lead page on triple canonization), card digests, heading and file listings, and the head of "Durable reports and current standing". The Other relations review read the page's Outcomes and Problem 1218 sections, two sections of the Problem 501 page with that page's own standing sentence, a section of the Problem 1171 page and a heading list. Content test: nothing in any report could only have come from the exposed text. Every finding carries its own witness on the frozen page, on a page image or in the reviewer's derivation: the Problem 1219 review's F1 to F4 rest on an Easton pattern, a ZFC coloring, the re-derivation of (T3) and the case $\lambda=\omega$; the Problem 1220 review's F1 and F2 on the constant coloring and the page image of p. 419; the Conjecture 1A review's F1 and F3 on explicit colorings and cardinal arithmetic; the Problem 1167 review's F1 on its own Lemma B; the Other relations review's F2 on the coset partition of the reals. The direction of every attack follows from the subject text, and none of the exposed text is a review of, or a verdict on, the reviewed sections. The exposures are ruled immaterial for all seven reports.

Reports graded

Tools review: pass. The subject block resolves (subject date, path, section, lines 39--108, matching the H2 lines). The independence facts and five disclosed over-reads are stated. The restatement S1 to S5 carries the convention, the arity, the growth data and the hypothesis (H) with its four quantifiers, and the universal quantifiers of (T2) and (T3) over $\lambda$, $\gamma$, $\rho_\nu$ and the witness. All ten checklist items carry an explicit verdict by name, with the inapplicable ones marked. The three weakest steps are re-derived: the growth condition and $2^{2+\kappa}<\lambda_0$ from S1's hypotheses alone, the identification $\chi=\sup_i2^{\mu(i)}=\sum_{i<\kappa}2^{\mu(i)}$ with the small-$J$ bound, and the color $1+\nu$ case of (T2); the grader re-derived each and agrees. The strongest attack is real and filed per statement: a hypothesis of the lemma that S1 might not supply, a misalignment of colors for infinite $\gamma$, the survey's remark read for a conditional or a reversed arrow on the page image, and the concentration and finite-first-term attacks on (T3). The premises carry their interfaces and reading depth. The verdict is stated in full per statement and assigns no tier.

Problem 1219 review: pass. The subject block resolves (lines 109--169). The independence facts and exposures are stated, with the excluded Known Results named where a finding depends on their wording. The restatement S0 to S10 carries the theorem page's conventions and every quantifier. All ten checklist items carry an explicit verdict by name. The three weakest steps are re-derived: the three left-to-right cases of the dividing line with $\chi$ identified in each, (T3) without its non-constancy hypothesis, and the step from the $\aleph_{\omega+\omega}$ sentence to the sentence about the site's bullet; the grader re-derived each and agrees. The strongest attack is real and succeeds twice: the ZFC coloring giving $\sum_k2^{\aleph_{n_k}}\not\to(\aleph_{\omega+1})^2_2$ (Sierpiński colorings on blocks of size $2^{\aleph_{n_k}}$, one color across blocks), checked here, and the superfluity of the non-constancy hypothesis in (T3), checked here; the Easton witness for the strong reading of the $\aleph_{\omega+\omega}$ sentence is a legitimate Easton pattern and the three conditions of Theorem 1.2 hold in it whatever $2^{\aleph_\omega}$ is. The premises carry interfaces and reading depth. The verdict is stated in full per statement and assigns no tier.

Problem 1220 review: pass. The subject block resolves (lines 171--285). The independence facts and three exposures are stated. The restatement S1 to S8 separates the two sentences that fall (S1b, S6b) from the statements around them and carries the side condition $\theta\le\kappa$ and the strong limit hypothesis. All ten checklist items carry an explicit verdict by name. The three weakest steps are re-derived: the realizability (H) through (ER) with $\mu(\alpha)^+\ge\kappa\ge\theta$, the sizes $\sum_i\lambda_i=\lambda$ and $|\bigcup_{\alpha\in I}B_\alpha|=\lambda$, and the obstruction re-derived from the lemma's growth condition rather than from the type count; the grader re-derived each and agrees. The strongest attack is real: each reported item against the page images of pp. 419--420 (which found S1b), both hypotheses at $\aleph_{\mathfrak c^+}$ in the [170] model, four places where the transfer lemma could lack a hypothesis, the constant coloring against S6b, and the color-merging induction. The premises carry interfaces and reading depth; the reviewer's Claim R, a proof of $\kappa\to(\kappa,\aleph_1)^2$ for every regular $\aleph_0$-inaccessible $\kappa$ by an elementary submodel closed under countable sequences whose intersection with $\kappa$ is an ordinal, and a recursion of length $\omega_1$, was checked here step by step and is correct; Theorem 7(i) of the held 1956 paper reads on the text layer of printed p. 432 as the report quotes it. The verdict is stated in full per statement and assigns no tier.

Conjecture 1A review: pass. The subject block resolves (lines 286--445 and the whole lead page). No exposure occurred. The restatement S1 to S13 carries the hypothesis H, the blocks, the pattern convention, the canonization statement (C) with its three clauses quantified, and a map of the lead's sentences onto the section's. All ten checklist items carry an explicit verdict by name. The three weakest steps are re-derived: the split of (P3) into the constancy of $d_j$ and the constancy of its value on each lower block, with a witness that the first does not give the second; the (P1) fiber argument with the thinning confined to $S_i$; and the endgame lemma with its exhaustive case split. The grader re-derived each and agrees, including that the endgame uses (P3) only for points of $S_i$ and that the triples $T_i$ cannot be thinned. The strongest attack is real and filed per statement, with two successes: the parenthetical of S8a against the witness $2^{\aleph_0}=\aleph_2$, $\mu(0)=\aleph_2$, $|C'|=\aleph_3=(2^{\aleph_0})^+$, checked here, and S9b against the coloring $f(\{a,b,b'\})=\varphi(a)$. The premises carry interfaces, reading depth, the imported relations flagged as not held, and the route's explicit assumptions. The verdict is stated in full per statement, states the lead's inheritance, and assigns no tier.

Problem 1218 review: pass. The subject block resolves (lines 446--510). The independence facts and exposures are stated. The restatement S1 to S11 names the two forms (P) and (S) and the premise $R$ and carries GCH at every sentence that needs it. All ten checklist items carry an explicit verdict by name. The three weakest steps are re-derived: the lift of $R$ through (T2) in the instance, with $\operatorname{cf}\aleph_\beta$ computed; the identification with the stepping-up instance under cardinal addition and the two conditionals; and the obstruction arithmetic with $\chi=\aleph_\beta$ under GCH; the grader re-derived each and agrees. The strongest attack is real and filed per statement, from the subscript and target count against the survey's Problem 1 to the consistency reading of a positive answer. The premises carry interfaces and reading depth, with [45] known only through the survey's sentence. The verdict is stated in full per statement and assigns no tier.

Problem 1167 review: pass. The subject block resolves (lines 511--540). The independence facts and six exposures are stated. The restatement S1 to S8 carries the cardinal-addition convention and the conjecture's chain hypothesis. All ten checklist items carry an explicit verdict by name. The three weakest steps are re-derived: the instance of the implication with its conclusion, proved outright by the reviewer's Lemma A (no monotone $\kappa^+$-sequence in the lexicographic order of ${}^\kappa2$), Lemma B ($2^\kappa\not\to(\kappa^+,4)^3$) and Corollary C, with Observation D settling every two-color and finite-target instance of the named case; the compatibility remark through the chain hypothesis; and the obstruction, strengthened by $\operatorname{cf}\chi=\kappa$, which keeps $\chi$ from being any power $2^\nu$. The grader re-derived Lemmas A and B, Corollary C and Observation D and agrees. The strongest attack is real: fidelity against a 300 dots per inch crop of Problem 2, the routing of the mechanism's conclusions onto a cardinal $2^{\lambda'}$, the parameters of the instance against both forms of the problem, and the support of the closing clause. The premises carry interfaces and reading depth. The verdict is stated in full per statement and assigns no tier.

Other relations review: pass. The subject block resolves (lines 542--567). The independence facts and five exposures are stated. The restatement S1 to S7 fixes the mechanism's three features and compares every problem with its Statement paragraph. All ten checklist items carry an explicit verdict by name. The three weakest steps are re-derived: the size floor $\lambda_0>2^\kappa\ge2^{\aleph_0}$ and $\chi>2^{\mu(0)}\ge2^{\aleph_0}$; the absence of any relation on the page at a successor resource and (T2)'s interface at $\aleph_\beta$; and the block shape at $\aleph_\omega$ for Problem 623 with the clause argument that no conclusion of Lemma 1.1 delivers a free set; the grader re-derived each and agrees. The strongest attack is real and filed per statement, with the coset partition of the reals against the recorded reason for Problem 70. The premises carry interfaces and reading depth. The verdict is stated in full per statement and assigns no tier.

Corrections

C1. Page: wiki/research/erdos_1219/connections.md. Location: section "Problem 1219 beyond the catalog question", paragraph "What transfers", the clause "the relation holds exactly when the powers $2^{\aleph_{\omega+n}}$ are unbounded below $\chi$ and exceed $\lambda$". Replace it with: "the relation holds exactly when the sequence $\langle2^{\aleph_{\omega+n}}:n<\omega\rangle$ is not eventually constant and some $2^{\aleph_{\omega+n}}$ is at least $\lambda$ (then above it, since $\operatorname{cf}2^\mu>\mu\ge\aleph_0=\operatorname{cf}\lambda$)". Basis: the quantifier on "exceed" is unstated; read as "every $n$" the biconditional is false, since with $2^{\aleph_{\omega+n}}=\aleph_{\omega+\omega+n+1}$ for $1\le n<\omega$ (an Easton pattern) and $2^{\aleph_\omega}<\lambda$ the three conditions of Theorem 1.2 hold at $\lambda=\aleph_{\omega+\omega}$. The tail is cofinal, so its non-constancy is the sequence's, and some $2^\mu\ge\lambda$ with $\mu<\lambda$ forces some $2^{\aleph_{\omega+n}}\ge\lambda$ by monotonicity. The Problem 1219 review filed this as F1 at severity suggested; accepted on the grader's own verification, because the sentence is a claim surface with a reading under which it is false.

C2. Page: wiki/research/erdos_1219/connections.md. Location: the same paragraph, the clause from "so the site's "no for $\kappa>\omega$" cannot be read as a statement about every $\kappa>\omega$ in the form the catalog prints it, and the source it attributes it to ([ErHa71], p. 20, not held) should be read before that bullet is sharpened." Replace it with: "in the form the catalog prints the problem, with $\aleph_\kappa$ in place of $\aleph_\omega$, the sum $\sum_k2^{\aleph_{n_k}}$ keeps its integer indices and the site's bullet is a theorem of ZFC on both sides: for $\kappa>\omega$, blocks of size $2^{\aleph_{n_k}}$ carrying Sierpiński colorings, with one color across blocks, admit no homogeneous set of size $\aleph_{\omega+1}$ (a homogeneous set in the across-block color meets each block in at most $\aleph_{n_k}$ points, so has size at most $\aleph_\omega$; one in the other color lies inside one block), so $\sum_k2^{\aleph_{n_k}}\not\to(\aleph_{\omega+1})^2_2$ whatever the powers are; for $\kappa<\omega$, the relation (ER) of the theorem page at a block with $n_k\ge\kappa$ gives $(2^{\aleph_{n_k}})^+\to(\aleph_{n_k+1})^2_2$ with $\aleph_{n_k+1}>\aleph_\kappa$ and $(2^{\aleph_{n_k}})^+\le\sum_k2^{\aleph_{n_k}}$. The characterization above concerns the different sum $\sum_{\mu<\aleph_\kappa}2^\mu$, for which the answer at $\kappa=\omega+\omega$ is yes under consistent cardinal arithmetic (for instance $2^{\aleph_{\omega+n}}=\aleph_{\omega+\omega+n+1}$ for $1\le n<\omega$, an Easton pattern) and no in ZFC at $\kappa=\omega_1$; the source the site attributes the bullet to ([ErHa71], p. 20, not held) should be read before the bullet is recorded in either form." Basis: the sentence passes from the $\aleph_{\omega+\omega}$ instance of the characterization, which is about $\sum_{\mu<\aleph_{\omega+\omega}}2^\mu$, to the catalog's form, whose sum has integer indices and is unchanged by the substitution; for that sum the bullet holds in ZFC by the coloring above, checked here, and the "yes for $\kappa<\omega$" half, meaningless for the re-indexed sum, holds by (ER). The Problem 1219 review filed this as F2 at severity required; accepted.

C3. Page: wiki/research/erdos_1219/connections.md. Location: the same section, paragraph "Falsification checks", the sentence "(T3) needs the non-constancy to place blocks of size $2^{\mu(i)}$ inside $\chi$ with sum $\chi$; when the powers are eventually constant the same conclusion comes from Sierpiński alone, so nothing is lost." Replace it with: "(T3) carries the non-constancy hypothesis only to match Theorem 1.2's: its proof takes the $\mu(i)$ from Step 1 but uses nothing about the powers beyond $\sum_{i<\kappa}2^{\mu(i)}=\chi$, which the sum formula gives for any strictly increasing cofinal $\langle\mu(i)\rangle$ of infinite cardinals, since $\sup_i2^{\mu(i)}=\sup_{\mu<\lambda}2^\mu=\chi>\kappa$; so its conclusion holds for every singular $\lambda$ with $\operatorname{cf}\lambda\not\to(\operatorname{cf}\lambda)^2_2$, and when the powers are eventually constant it also comes from Sierpiński alone, so the case split loses nothing." Basis: the strict increase of the powers is used nowhere in (T3)'s proof; the necessity claim is false. The statement of (T3) keeps its hypothesis, as reviewed. The Problem 1219 review filed this as F3 at severity suggested; accepted on the grader's own verification, because the sentence is false as written.

C4. Page: wiki/research/erdos_1219/connections.md. Location: the same paragraph, "for regular $\lambda$ the relation reduces to $\lambda$ weakly compact and is not touched by the mechanism". Replace it with: "for uncountable regular $\lambda$ the relation reduces to $\lambda$ weakly compact (at $\lambda=\omega$ it is Ramsey's theorem) and is not touched by the mechanism". Basis: $\sum_{n<\omega}2^n=\aleph_0\to(\aleph_0)^2_2$ while $\omega$ is not weakly compact under the standard convention; for uncountable regular $\lambda$ the reduction holds (successor: Sierpiński; weakly compact: strongly inaccessible, $\chi=\lambda$; regular limit not weakly compact: $\chi=\lambda$, Sierpiński, or the (T3) argument with $\kappa=\lambda$). The Problem 1219 review filed this as F4 at severity note; accepted on the grader's own verification, because the sentence is false at its boundary.

C5. Page: wiki/research/erdos_1219/connections.md. Location: section "Problem 1220: the singular unbalanced relation", paragraph "Target", "Komjáth's commentary (p. 419, read in the held conversion) records:". Replace "(p. 419, read in the held conversion)" with "(pp. 419--420, read in the held conversion)". Basis: on the held PDF the sentences on [170] and [171] are printed on p. 420 (PDF p. 3), the rest of the list on p. 419. The Problem 1220 review filed this as F4 at severity suggested; accepted, a locator is a fidelity item.

C6. Page: wiki/research/erdos_1219/connections.md. Four locations. (a) The same Target paragraph: replace "gives the positive relation when $\lambda$ is strong limit;" with "gives the positive relation when $\aleph_{\mathfrak c^+}$ is strong limit;". (b) Paragraph "What transfers: the strong-limit case", the sentence "so the transfer lemma reproves the strong-limit case of the problem, the case Komjáth attributes to [45, Theorem 35.4]. Nothing new is claimed for it." Replace it with: "so the transfer lemma proves the strong-limit case of the problem, whose instance $\lambda=\aleph_{\mathfrak c^+}$ Komjáth attributes to [45, Theorem 35.4]; [45] is not held, so how much of the case its theorem covers is not checked here, and nothing new is claimed." (c) The section's Outcome paragraph: replace "Transfers to the strong-limit case, which is a known theorem reproved through the lemma;" with "Transfers to the strong-limit case, proved through the lemma from the Erdős--Rado theorem for pairs at the cofinality;". (d) Section "Outcomes", the Problem 1220 bullet: replace "transfers to the strong-limit case, a known theorem reproved (the transfer lemma:" with "transfers to the strong-limit case, proved from the Erdős--Rado theorem for pairs at the cofinality (the transfer lemma:". Basis, checked on the text layer of printed p. 419: the survey's sentence is "An Erdős--Rado theorem ([45, Theorem 35.4]) states that the positive relation holds if $\aleph_{\mathfrak c^+}$ is strong limit", an instance; the page reports it as the general strong-limit case, which is a strengthening of the source. The Problem 1220 review filed this as F2 at severity required; accepted, with the two outcome sentences added because they carry the same characterization.

C7. Page: wiki/research/erdos_1219/connections.md. Location: the same paragraph "What transfers: the strong-limit case", "the Erdős--Rado theorem for pairs (not held; the regular case is not what the problem asks)". Replace the parenthesis with: "(for successor $\kappa=a^+$ it is Theorem 7(i) of the held Erdős and Rado (1956), printed p. 432, PDF p. 6: $a^+\to(b,a^+)^2$ for the least $b$ with $a^b>a$, and $a^{\aleph_0}=a$ forces $b\ge\aleph_1$; the regular limit case is not held; the regular case is not what the problem asks)". Basis, checked on the text layer of printed p. 432: Theorem 7 reads "If $a\ge\aleph_0$, and if $b$ is minimal such that $a^b>a$, then (i) $a^+\to(b,a^+)^2$"; $\aleph_0$-inaccessibility of $a^+$ gives $a^{\aleph_0}=a$, so $b\ge\aleph_1$, and exchanging the colors gives $a^+\to(a^+,\aleph_1)^2$, which covers the simplest case $\operatorname{cf}\lambda=\mathfrak c^+$. The Problem 1220 review filed this as F5 at severity suggested; accepted, because a consumed premise's held source is a premise item.

C8. Page: wiki/research/erdos_1219/connections.md. Location: paragraph "What does not transfer, and why", two sentences. Replace "and no block inside $\lambda$ can be thinned to a single type." with "and the count that the second thinning of the lemma page uses no longer yields a fiber of size $\lambda_\alpha$." Replace "This is an obstruction to canonization over full parameter sets, not a proof of the negative relation;" with "This is an obstruction to the counting of Lemma 1.1 over full parameter sets, not to canonization as such and not a proof of the negative relation;". Basis: bound (A) is an upper bound on the number of types; for a particular $f$ a fiber of full size may exist (for the constant coloring every block is a single type over every parameter set), and what fails when $2^{|E_\alpha|}\ge\lambda$ is the pigeonhole of the second thinning, which is what the paragraph argues. The Problem 1220 review filed this as F1 at severity required; accepted.

C9. Pages: wiki/research/erdos_1219/connections.md and wiki/research/leads/triple_canonization_for_hajnal_conjecture/_index.md. Seven locations. (a) Connections page, section "Hajnal's Conjecture 1A: triples", the definition of (C), bullet (P3): replace "for all $a\in B_i$ and $\{b,b'\}\in[B_j]^2$" with "for all $a\in S_i$ and $\{b,b'\}\in[B_j]^2$". (b) The proof of the endgame lemma: replace "If $\eta_1=1$ and $\varepsilon=0$: one point of $B_i$ together with $T_j$, by (P3) at the pairs of $T_j$." with "If $\eta_1=1$ and $\varepsilon=0$: one point of $S_i$ together with $T_j$, by (P3) at the pairs of $T_j$." (c) Paragraph "What the lemma gives toward (C)", the (P3) bullet: replace "(P3) says that $d_j(\{b,b'\})=\langle f(\{a,b,b'\}):a\in E_j\rangle$, a coloring of $[B_j]^2$ with at most $2^{|E_j|}=2^{\mu(j-1)}$ colors, is constant." with "(P3) implies that $d_j(\{b,b'\})=\langle f(\{a,b,b'\}):a\in E_j\rangle$, a coloring of $[B_j]^2$ with at most $2^{|E_j|}=2^{\mu(j-1)}$ colors, is constant, and asks in addition that its common value $v_j:E_j\to2$ be constant on each $S_i$, $i<j$. Constancy of $d_j$ alone does not give this: if $f(\{a,b,b'\})=\varphi(a)$ on every triple of pattern $(i,j,j)$ with $\varphi$ not constant on $S_i$, then $d_j$ is constant for every choice of $B_j$ while (P3) fails. Once every $d_j$ is constant, the second demand follows by thinning each $S_i$ to a fiber of $a\mapsto\langle v_j(a):j>i\rangle$, a map with at most $2^{\aleph_0}$ values, taken jointly with the fiber of (P1) under the same hypothesis $2^{\aleph_0}<\mu(i)$; the values $v_j(a)$ do not change when the $S_j$ are thinned later." (d) Paragraph "Next investigation": replace "A positive outcome, with the block statement for (P2), proves the conjecture under $2^{\aleph_0}<\aleph_\omega$ by the endgame lemma;" with "A positive outcome, with the block statement for (P2) and the fiber step of the (P3) bullet that makes the common value of each $d_j$ constant on the lower $S_i$, proves the conjecture under $2^{\aleph_0}<\aleph_\omega$ by the endgame lemma;". (e) Lead page, "The canonization statement (C)", bullet (P3): replace "with one point in $B_i$ and two in $B_j$." with "with one point in $S_i$ and two in $B_j$." (f) Lead page, section "The consequential gap", after "What is open is whether the particular colorings induced by an $f$ without color-$1$ $4$-sets are better than arbitrary ones.", add: "Constancy of the induced coloring on $[B_j]^2$ leaves a value that may still depend on the lower point; under $2^{\aleph_0}<\aleph_\omega$ that dependence is removed by thinning each $S_i$ to a fiber of the map sending $a$ to its values against the higher blocks, at most $2^{\aleph_0}$ of them, jointly with the fiber behind (P1)." (g) Lead page, section "Next step", item 1: replace "A positive outcome, with the block statement behind (P2), proves the conjecture under $2^{\aleph_0}<\aleph_\omega$ by the endgame lemma." with "A positive outcome, with the block statement behind (P2) and the fiber step of the consequential gap, proves the conjecture under $2^{\aleph_0}<\aleph_\omega$ by the endgame lemma." Basis, checked here: (P3) at $(i,j)$ is the conjunction of the constancy of $d_j$ on $[B_j]^2$ and the constancy of its value on $B_i$; the witness coloring separates them; the endgame lemma's proof uses (P3) only with the single lower point taken in $S_i$ (case $\eta_1=1$) or with all points in the $S_i$ (final case), so (C) with (P3) over $S_i$ still supports it; the three fixed points of $T_i$ are chosen at stage $i$ against one point of each higher block and cannot be thinned, so (P3) over $B_i$ is not deliverable; and for $a\in S_i$ the map $a\mapsto\langle v_j(a):j>i\rangle$ takes at most $2^{\aleph_0}$ values, so under $2^{\aleph_0}<\mu(i)$ (regular, as every $\mu(i)$ is an $\aleph_m$ with $m<\omega$) a fiber of size $\mu(i)$ exists, can be taken jointly with the (P1) fiber, and is unaffected by later thinning of the $S_j$, whose common values $v_j$ restrict. With these changes the positive-outcome conditional follows from the endgame lemma. The Conjecture 1A review filed this as F1 at severity required with two alternative repairs; accepted in the first form, which keeps the (P3) question as the section poses it.

C10. Page: wiki/research/erdos_1219/connections.md. Location: the same section, the (P2) bullet, "inside which the relation $(2^\mu)^+\to(\mu^+,4)^3$ cannot be applied again ($|C'|$ is not of that form),". Replace it with: "inside which the relation $(2^\mu)^+\to(\mu^+,4)^3$ helps no further (it applies only when $\mu(i)=2^\nu$ for some $\nu$, and then $2^\nu<\aleph_\omega\le2^{\mu_0}$ gives $\nu^+\le\mu_0<\mu(i)$, so it yields a color-$0$ set of size $\nu^+<\mu(i)$ only),". Basis: under $2^{\aleph_0}=\aleph_2$, $n(0)=1$ and $2^{\aleph_2}>2^{\aleph_1}$, Step 1 gives $\mu(0)=\aleph_2$, the sentence's own hypothesis $2^{\aleph_0}\le\mu(0)$ holds, and $|C'|=\aleph_3=(2^{\aleph_0})^+$ is of the form the parenthesis denies; the consequence survives because the set the relation yields has size $\nu^+<\mu(i)$. The Conjecture 1A review filed this as F3 at severity suggested; accepted on the grader's own verification, because the parenthesis is false as written.

C11. Pages: wiki/research/erdos_1219/connections.md and the lead page. (a) Connections page, paragraph "Next investigation": replace "shows that a proof of the conjecture cannot pass through canonization over the lower blocks, and its $f$ is a candidate" with "shows that (C) fails for this $f$ on these blocks, so the route as formulated is closed, while a weaker canonical form or another decomposition is not excluded; its $f$ is a candidate". (b) Lead page, "Next step", item 1: replace "A negative outcome shows that no proof passes through canonization over the lower blocks and supplies a coloring to test directly against the conjecture." with "A negative outcome shows that (C) fails for that coloring on these blocks, closing the route as formulated without excluding a weaker canonical form or another decomposition, and supplies a coloring to test directly against the conjecture." Basis: a negative outcome refutes the realizability of the property carrying (P3), hence (C) for that $f$ and those blocks; the universal statement about proofs has no argument. The Conjecture 1A review filed this as F2 at severity suggested; accepted on the grader's own verification, because a consequence sentence is a claim surface and this one is unsupported.

C12. Page: wiki/research/erdos_1219/connections.md. Location: section "Problem 1218: the negative triple relation under GCH", paragraph "Context recorded: the stepping-up instance", "Komjáth's commentary on Problem 5 (p. 419)". Replace "(p. 419)" with "(p. 420)". Basis: on the held PDF Problem 5 and its commentary are printed on p. 420 (PDF p. 3). The Problem 1218 review filed this as F1 at severity suggested; accepted, a locator is a fidelity item.

C13. Page: wiki/research/erdos_1219/connections.md. Location: section "Problem 1167: the negative stepping-up lemma", paragraph "What transfers", "so the implication asked would give $2^{\aleph_n}\not\to(\aleph_\omega,4)^3$ for every $n$, while Conjecture 1A asserts the positive relation for the sum $\chi$ of these powers;". Replace it with: "so the implication asked would give $2^{\aleph_n}\not\to(\aleph_\omega,4)^3$ for every $n$; that relation holds outright, from $2^{\aleph_n}\not\to(\aleph_{n+1},4)^3$ (author-recorded; a theorem of Erdős, Hajnal and Rado 1965, not held: order ${}^{\aleph_n}2$ lexicographically, fix a well-order of it, and color a triple $1$ exactly when its lexicographic middle element is the well-order-least of the three; in a color-$1$ four-element set the two middle elements would each precede the other, and in a color-$0$ set of size $\aleph_{n+1}$ the elements with at most $\aleph_n$ elements of the set below them, and likewise above, number at most $\aleph_n$ each, since the lexicographic order carries no monotone sequence of length $\aleph_{n+1}$, so the well-order-least of the remaining elements has elements of the set on both sides; the derivation is written out as Lemma B of the section's review), so this instance of the implication is settled, while Conjecture 1A asserts the positive relation for the sum $\chi$ of these powers;". Basis: as written, the closing clause "a proof of the conjecture would show that the sum of the powers behaves differently from each summand" consumes the summand relations, which the text supplies only as what the open implication "would give", so the clause silently depends on Problem 1167; the summand relations are theorems, by the reviewer's Lemmas A and B, which the grader re-derived. The Problem 1167 review filed this as F1 at severity required; accepted, with the exact text supplied here.

C14. Page: wiki/research/erdos_1219/connections.md. Location: section "Other relations in the set-theory folder", "nor to Problem 70, a relation for triples on the order type of the reals, which has no block decomposition into regular cardinals;". Replace it with: "nor to Problem 70, a relation for triples with an ordinal target on the reals: (T1) passes to the lemma the hypothesis $2^{\chi+\kappa}<\lambda_0\le|A|$, which puts every domain of the lemma above $2^{\aleph_0}=|\mathbb R|$, and Theorem 1.2's resource exceeds $2^{\mu(0)}\ge2^{\aleph_0}$, so no coloring of a set of size continuum is in the mechanism's scope, while (T2) and (T3) produce negative relations for pairs and bear on no positive relation for triples;". Basis: by the page's Scope paragraph a block decomposition is any partition into $\kappa$ pieces, and $\mathbb R=\bigsqcup_{x\in\mathbb R/\mathbb Q}(x+\mathbb Q)$ is a partition into countable, hence regular-sized, pieces, so the recorded reason is false as written; the size floor is the operative reason and blocks both readings of "the ordinal of the real numbers". The Other relations review filed this as F2 at severity required; accepted.

Rejected and downgraded findings

• Tools review, F1 (suggested: say in (T3)'s choice sentence that the $\mu(i)$ are at least $\kappa$). Downgraded to optional; no change required. The sentence invokes Step 1 of the theorem page, whose $\rho_i\ge\kappa$ returns $\mu(i)>\kappa$, infinite, so (S) applies to every block; the phrase may be added.
• Tools review, F2 (note: "since $\lambda$ is a limit cardinal" is not the operative reason). Downgraded to optional; no change required. The product $|i_0|\cdot\mu(i_0)$ is the larger of two cardinals below $\lambda$; the stated reason is superfluous and may be dropped.
• Tools review, F3 (note: $(\rho_\nu)_{\nu<\gamma}$ is undefined on the page) and F4 (note: $n\ge1$). Downgraded to optional; no change required. The proof of (T2) fixes the reading, which is the survey's notation, and the lemma page's $1\le n_i$ governs (T1); both clauses may be added.
• Problem 1219 review, F5 (note: "two classical facts" undercounts). Rejected; no change. The display uses exactly two, the block coloring and Sierpiński's theorem; the weakly compact characterization enters only the reformulation sentence, which names it.
• Problem 1220 review, F3 (suggested: the strong-limit reason for non-constancy). Downgraded to optional; no change required. The reason is incomplete rather than wrong: inside the transfer lemma $\lambda$ is strong limit, and an eventually constant value $c$ with $c\ge\mu^+$ for all large $\mu<\lambda$ would satisfy $c\ge\lambda$; the proposed wording may be adopted.
• Problem 1220 review, F6 (note: item (i) asks a question the reported theorem answers), F7 (note: "must" and "would need" are programmatic) and F8 (note: two attributions outside the read set). Downgraded to optional; no change required. F6 and F7 are right and the softer wording may be adopted. For F8 the grader checked the Problem 1171 page: its "Other relations" section says that its (T3) would give $\lambda\to(\lambda,(\aleph_1)_k)^2$ from the relation asked, "a corollary if the problem is settled positively and no help toward settling it", as the section reports; the catalog page is excluded.
• Conjecture 1A review, F4 (note: clause (3)'s heredity hypothesis is not met as stated). Downgraded to optional; no change required. The page invokes the argument of clause (3), and the reviewer's own W2 shows the fiber step with the thinning confined to $S_i$ is sound; the proposed wording may be adopted on both pages.
• Conjecture 1A review, F5 (note: $f_0$ lies outside the endgame lemma's hypotheses), F6 (note: the first thinning is outside (T1)), F7 (note: (BS2) is stronger than the route needs; the (P3) question depends on the lower sequence) and F8 (note: the Outcome sentence omits the imported triple relation). Downgraded to optional; no change required. Each observation is right: $f_0$ makes every block homogeneous in color $0$; the first thinning is cited by name from the lemma page; "needs" may read "is supplied by"; the triple relation is carried in the section's next-investigation paragraph and in the lead's premises.
• Problem 1218 review, F2 (suggested: a gloss on the catalog's $\operatorname{cf}$ and $\operatorname{cr}$). Downgraded to optional; no change required. The sentence is reportage of a page outside every read set, marked "the page does not define $\operatorname{cr}$"; the reviewer's index reading is coherent (the grader confirms $\operatorname{cf}\aleph_{\omega_{\omega+1}}=\aleph_{\omega+1}$, index $\omega+1$, predecessor $\omega$, and $\operatorname{cf}\aleph_\omega=\aleph_0$ for the count of fours) but rests on an unread page; the gloss may be added.
• Problem 1218 review, F3 (note: finite $\mu$; a stronger obstruction), F4 (note: name [45]), F5 (note: unverified sentences), F6 (note: the labels of the two forms) and F7 (note: the survey's Problem 2 prints $\gamma\le2$). No change required. F3's stronger form is right (under GCH $\kappa=\aleph_{\omega+1}=2^{\aleph_\omega}\not\to(\aleph_{\omega+1})^2_2$ as well); F7 is confirmed on the page image at 300 dots per inch, and the section cites the corpus statement, which carries no such bound.
• Problem 1167 review, F2 (suggested: say that the two-color instances of the named case are settled). Downgraded to optional; no change required beyond C13. Observation D is verified: for $r=2$, finitely many colors and finite targets at least $3$ in the colors after the first, the consequent holds when $\kappa_0\le\lambda$ by (EDM) and Ramsey's theorem, and the antecedent fails when $\kappa_0>\lambda$ by Lemma B; the sentence may be added.
• Problem 1167 review, F3 (note: the survey's Problem 2 is the contrapositive with side conditions), F5 (note: the compatibility rests on the chain hypothesis) and F6 (note: catalog attributions unchecked). Downgraded to optional; no change required. The page image shows "$\gamma\le2$, $\kappa_\nu>r$" in Problem 2, so "prints it as Problem 2 with the same case list" is loose and a clause may be added; F5 is right and one clause may name the reason.
• Problem 1167 review, F4 (note, outside the section: the catalog form of Problem 1167 without $\kappa_\alpha>r$ fails at degenerate parameters). Verified by the grader: with $r=2$, $\gamma=2$, $\lambda=\aleph_0$, $\kappa_0=\aleph_1$, $\kappa_1=2$, every coloring of triples either gives some triple the color $1$ or is constantly $0$, so $2^{\aleph_0}\to(\aleph_1,3)^3$ holds, while the constant coloring refutes $\aleph_0\to(\aleph_1,2)^2$. It concerns the problem page, not the section; recorded for the problem page's next revision.
• Other relations review, F1 (suggested: "beyond the GCH remark" reads as a use of the tools on Problem 1168), F3 (note: "balanced or" is not an obstruction for this mechanism), F4 (note: the Problem 501 page's obstructions are to its own tools; $\omega_{\omega+1}$ is regular) and F5 (note: "disjoint in what they consume"). Downgraded to optional; no change required. The sentence on Problem 1168 already says the remark uses (T2) at $\aleph_\beta$ and not at $\aleph_{\omega+1}$; the operative reason for Problems 1170 and 1172 is stated ("on the regular cardinals"); the more precise wordings may be adopted.
• Other relations review, F6 (note, outside the section: under GCH the first relation of Problem 1172 as the problem page prints it follows from (ER)). Verified by the grader: (ER) at $\mu=\aleph_1$ under GCH is $\aleph_3\to(\aleph_3,\aleph_2)^2$, and homogeneous sets of sizes $\aleph_3$ and $\aleph_2$ contain subsets of order types $\omega_2$ and $\omega_1+2$. It concerns the problem page, at the standing of (ER) on the theorem page; recorded for that page's status search.

Checks of the grader's own that produced no correction. The seven line ranges the reports name match the H2 lines of the frozen page. On the held Komjáth PDF, printed p. $n$ is PDF p. $n-417$; p. 419 carries Problems 1 to 4 with the necessity remark under Problem 4 and the [45, Theorem 35.4] sentence, and p. 420 the [170] and [171] sentences and Problem 5 with its commentary; Problem 2 prints "$\gamma\le2$, $\kappa_\nu>r$". The Known Results bullet of Problem 1219 carries the site remark exactly as the section quotes it, "With $\aleph_\kappa$ in place of $\aleph_\omega$ the answer is yes for $\kappa<\omega$ and no for $\kappa>\omega$", attributed to [ErHa71, p. 20], not held, not read. The Problem 1218 section's premise $R$, the identification of (S) as the $r=2$ instance with countably many targets $3$, and the direction (S) implies (P) were re-derived here. The transfer lemma of the Problem 1220 section, the endgame lemma and the (P1) fiber of the Conjecture 1A section, and the coset partition of the reals were re-derived here as well.

Graded verdicts

Tools. S1 (T1, the lemma's interface): refutation-failed. S2 (T2): refutation-failed. S3 (the attribution under T2): refutation-failed, confirmed on the page image by the reviewer and on the text layer here. S4 (T3): refutation-failed. S5: not a claim.

Problem 1219 beyond the catalog question. S0: not a claim. S1 (the dividing line): refutation-failed. S2 (the weakly compact reformulation): refutation-failed. S3 (the $\aleph_{\omega_1}$ instance): refutation-failed. S4 (the $\aleph_{\omega+\omega}$ instance): refutation-failed under the eventual reading, with its quantifier fixed by C1. S5 (the sentence on the site's bullet): refuted-as-stated; replaced by C2. S6 (the necessity of non-constancy in T3): refuted-as-stated; replaced by C3. S7 (coverage): refutation-failed. S8 (comparison with T2): refutation-failed. S9 (the regular case): refuted-as-stated at $\lambda=\omega$, refutation-failed for uncountable regular $\lambda$; fixed by C4. S10 (standing): refutation-failed.

Problem 1220: the singular unbalanced relation. S1 (fidelity to the survey): refutation-failed, except S1b (the general strong-limit case attributed to [45, Theorem 35.4]): refuted-as-stated; replaced by C6, with the locator fixed by C5. S2 (the conditional on [170]): refutation-failed. S3 (the transfer lemma): refutation-failed. S4 (the equivalence for strong limit $\lambda$): refutation-failed. S5 (the composition with the Erdős--Rado theorem for pairs): refutation-failed; the regular limit case rests on the reviewer's Claim R, checked here, and the successor case on the held 1956 paper by C7. S6a (the obstruction): refutation-failed in the form "the hypotheses of Lemma 1.1 can be met for a block decomposition of $\lambda$ with the union of the $B_\alpha$ as the homogeneous set exactly when $\lambda$ is strong limit". S6b ("no block inside $\lambda$ can be thinned to a single type"): refuted-as-stated; replaced by C8. S7 (the color-merging consequence): refutation-failed. S8: not a claim.

Hajnal's Conjecture 1A: triples. S1, S2: not claims. S3 (the endgame lemma): refutation-failed. S4 (the remark on the triples $T_i$): refutation-failed, with the scope of its clause (c) as the review narrows it. S5 (the application of T1 with the properties $P_i$): refutation-failed, conditional on the imported relation $(2^\mu)^+\to(\mu^+,4)^3$, not held. S6 ((P1) under $2^{\aleph_0}<\aleph_\omega$): refutation-failed. S7 (the reduction of (P2) to the constancy of $c_i$): refutation-failed. S8a ("cannot be applied again"): refuted-as-stated; replaced by C10. S8b (the rest of the Erdős--Rado discussion and the need for the block statement): refutation-failed. S9a (no clause of the lemma reaches (P3)): refutation-failed. S9b ("(P3) says that $d_j$ is constant"): refuted-as-stated; replaced by C9. S10 (the counting obstruction): refutation-failed. S11 (the falsification checks): refutation-failed. S12a (a positive outcome with the block statement proves the conjecture under $2^{\aleph_0}<\aleph_\omega$): proof defective; statement unresolved as printed, the bridge from the constancy of $d_j$ to (P3) being absent; with C9 applied the corrected conditional follows from the endgame lemma, as the review's W1 and this grade re-derive. S12b (a negative outcome excludes every proof through canonization over the lower blocks): proof defective; statement unresolved as printed; C11 replaces it with what the negative outcome shows. S13 (the outcome sentence): refutation-failed as author-recorded, with the imported triple relation among its premises.

Problem 1218: the negative triple relation under GCH. S1 (target): refutation-failed. S2 (the two forms): refutation-failed. S3 (the catalog's general conjecture): not a claim of the section, reportage. S4 (regular resource): refutation-failed. S5 (the sum of powers under GCH): refutation-failed. S6 (wrong exponent): refutation-failed. S7 (the stepping-up instance): refutation-failed. S8 (the premise $R$ under GCH): refutation-failed at the author-recorded standing of the imported Section 20 theorem, with the locator fixed by C12. S9, S10 (the two conditionals): refutation-failed. S11: not a claim.

Problem 1167: the negative stepping-up lemma. S1 (the restatement of the problem): refutation-failed. S2 (nothing transfers): refutation-failed. S3 (the trivial negative relation): refutation-failed. S4 (the conditional instance): refutation-failed; its conclusion is a theorem, so the instance is settled. S5 (compatibility): refutation-failed. S6 (a proof of the conjecture would separate the sum from its summands at exponent $3$): refutation-failed, its support completed by C13. S7 (Problem 1218 as the catalog instance): refutation-failed. S8: not a claim.

Other relations in the set-theory folder. S1 (the mechanism's three features): refutation-failed. S2 (Problem 1168): refutation-failed. S3 (Problem 70): refutation-failed for the non-applicability; its recorded reason refuted-as-stated and replaced by C14. S4 (Problems 1170 and 1172): refutation-failed. S5 (Problem 598): refutation-failed. S6 (Problems 623 and 1173): refutation-failed. S7 (the Problem 1171 page's induction): refutation-failed.

Lead: triple canonization for Hajnal's conjecture. Outcome: needs_update. Two statements the lead's route rests on fail as printed: the restatement of (P3) as the constancy of the induced pair coloring (S9b, carried in "The consequential gap" and in the (C) bullet that asks (P3) over $B_i$) is refuted-as-stated, and the positive-outcome conditional of its Next step 1 (S12a) is proof defective; statement unresolved as printed; the negative-outcome sentence of the same item (S12b) is proof defective; statement unresolved as well. C9 and C11 state the exact repair on the lead page and on the connections page. The endgame lemma, the (P1) transfer, the (P2) reduction and the (P3) counting obstruction, which the route otherwise rests on, are refutation-failed, and with C9 and C11 applied every statement the route rests on is refutation-failed, by the review's W1 and this grade's check; the lead stays author-recorded and unreviewed until its text is updated.

No tier is assigned and no status changes.