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Scope. This page carries the reconstructed mechanism of Lemma 2.1 (the color-merging induction) and Theorem 3.1 (the restriction from $\omega_1^2$ to its initial segment $\omega_1\omega$) to the other problems that use the same objects, and records for each what the transfer gives, what it does not give, and why. Every statement here is author-recorded, no problem status changes, and no tier is assigned. The statements were independently reviewed as they stood on 2026-09-28T13:50:48Z by eleven focused reviews graded in the connections grade, which passes all eleven and accepts sixteen corrections, since applied. Sources are cited at the standing their library pages record. Reading outside the held corpus is limited to the erdosproblems.com pages named in each section, and one arXiv abstract, named where it is used. Notation is that of the lemma page; a witness for a negative relation is a coloring that refutes it.

Tools

Five facts are used below. The first two are on the reconstruction pages, the third is stated on the folder index, and the last two are recorded here with their proofs.

(T1) Transport. Fact 1 of the lemma page is stated for ordinal targets; the same proof gives it for a list of targets mixing order types and graphs. Let $\gamma\to(\beta,H_1,\ldots,H_k)^2_{k+1}$ mean: every coloring of $[\gamma]^2$ with the colors $0,\ldots,k$ has a subset of order type $\beta$ homogeneous in color $0$, or, for some $i$, a copy of $H_i$ in color $i$, a copy being a subset of type $H_i$ all of whose pairs have color $i$ when $H_i$ is an order type, and a subgraph isomorphic to $H_i$ all of whose edges have color $i$ when $H_i$ is a graph. If $Y$ is a set of ordinals with $\operatorname{otp}(Y)=\gamma$, the relation holds for colorings of $[Y]^2$: an order isomorphism carries pairs to pairs, subsets of type $\beta$ to subsets of type $\beta$, and copies of $H_i$ to copies of $H_i$. Only order isomorphisms are used, so everything below also holds with the ordinals replaced by arbitrary order types, the setting of Problem 118.

(T2) Restriction and weakening. If $\gamma\to(\beta,H_1,\ldots,H_k)^2_{k+1}$ and $\gamma'$ has a subset of type $\gamma$ (for ordinals, $\gamma'\ge\gamma$), then $\gamma'\to(\beta,H_1,\ldots,H_k)^2_{k+1}$: restrict the coloring and read the result in $\gamma'$, as in Steps 2 and 3 of the theorem page. If $\beta'\le\beta$, then $\gamma\to(\beta',H_1,\ldots,H_k)^2_{k+1}$, since a set of type $\beta$ has a subset of type $\beta'$; a target $H_i$ may be replaced by any $H_i'$ that embeds in $H_i$; and a color may be left unused, so a relation with fewer triangle colors is weaker.

(T3) The induction with general targets. If $\alpha\to(\alpha,H_i)^2$ for $i=1,\ldots,k$, then $\alpha\to(\alpha,H_1,\ldots,H_k)^2_{k+1}$. This is the observation recorded on the folder index; it is the constant case $\gamma_0=\cdots=\gamma_k=\alpha$ of (T4).

(T4) Descent lemma. Let $k\ge1$, let $\gamma_0,\ldots,\gamma_k$ be ordinals and $H_1,\ldots,H_k$ targets, and suppose

$$\gamma_{j-1}\to(\gamma_j,H_j)^2\qquad(j=1,\ldots,k).$$

Then $\gamma_0\to(\gamma_k,H_1,\ldots,H_k)^2_{k+1}$, with color $j$ carrying the target $H_j$.

Proof. Induction on $k$; for $k=1$ the conclusion is the hypothesis. Assume the lemma for chains of length $k$ and let the $k+1$ relations hold for $\gamma_0,\ldots,\gamma_{k+1}$. Let $c:[\gamma_0]^2\to\{0,\ldots,k+1\}$ and define $c'$ by $c'(p)=0$ if $c(p)\in\{0,k+1\}$ and $c'(p)=c(p)$ otherwise. Then $c'$ has the colors $0,\ldots,k$, and the lemma for the first $k$ relations gives one of two things. Either there are $j\le k$ and a copy of $H_j$ all of whose pairs have color $j$ under $c'$, hence under $c$, which agrees with $c'$ on the colors $1,\ldots,k$. Or there is $Y\subseteq\gamma_0$ with $\operatorname{otp}(Y)=\gamma_k$ and $c(p)\in\{0,k+1\}$ for every $p\in[Y]^2$; then $\gamma_k\to(\gamma_{k+1},H_{k+1})^2$, transported to $Y$ by (T1) and applied to the two-coloring of $[Y]^2$ with the colors $0$ and $k+1$, gives a subset of $Y$ of type $\gamma_{k+1}$ homogeneous in color $0$ or a copy of $H_{k+1}$ homogeneous in color $k+1$. In every case the conclusion for $k+1$ holds. The lemma page merges the colors $0$ and $1$ and renames the rest; merging $0$ with the last color is the same induction with the colors relabeled (fact 2 of the lemma page). $\square$

(T5) Witness extraction. Suppose $c:[\alpha]^2\to\{0,\ldots,k\}$ witnesses $\alpha\not\to(\alpha,3,\ldots,3)^2_{k+1}$: no subset of type $\alpha$ is homogeneous in color $0$, and no triangle is homogeneous in a color $j\ge1$. Then there are $Y\subseteq\alpha$ with $\operatorname{otp}(Y)=\alpha$ and $j\in\{1,\ldots,k\}$ such that the two-coloring of $[Y]^2$ giving color $1$ to the pairs of $c$-color $j$ and color $0$ to all other pairs witnesses $\alpha\not\to(\alpha,3)^2$ after transport. Proof. Induction on $k$; for $k=1$ take $Y=\alpha$ and $j=1$. For $k\ge2$ merge the colors $0$ and $k$ of $c$ into a coloring $c'$ with the colors $0,\ldots,k-1$. If $c'$ has a subset $Y$ of type $\alpha$ homogeneous in color $0$, then $c$ takes only the values $0$ and $k$ on $[Y]^2$, and the two-coloring with $j=k$ has no color-$0$ subset of type $\alpha$, since $c$ has none, and no color-$1$ triangle. Otherwise $c'$ witnesses the relation with $k-1$ triangle colors, and the induction hypothesis gives $Y$ and $j\le k-1$; the two-colorings built from $c'$ and from $c$ with this $j$ coincide, since $c'$ agrees with $c$ on the colors $1,\ldots,k-1$. $\square$

Together with Lemma 2.1 and (T2), (T5) gives, for every order type $\alpha$ and every finite $k\ge1$,

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

and adds that a many-color witness contains a two-color witness on a subset of full type: adding triangle colors changes neither the positive nor the negative relation.

Problem 1171 in ZFC: descent chains and their obstruction

Target. Problem 1171 asks, for every finite $k$, whether $\omega_1^2\to(\omega_1\omega,3,\ldots,3)^2_{k+1}$ with $k$ triangle targets. Its current assessment records the status open: the relation holds under $\mathrm{MA}_{\aleph_1}$ for every $k$, so ZFC does not refute it; the cases $k\le2$ are theorems of ZFC, by the relation of Baumgartner and Hajnal for $k=2$ (the positive relation), which contains $k=1$ (Komjáth attributes $k=1$ to a 1970 paper of Erdős and Hajnal, where the problem page does not find it); and the case $k=3$ is recorded by Komjáth 2025 (Problem 54 discussion) as unknown. Bloom's catalog labeled the problem not disprovable, a change prompted by the deposit that this folder reconstructs; its author withdrew the deposit, as its claim page records, and the consistency rests on Baumgartner's theorem without it. The question the reconstructed route leaves open is the ZFC question for $k\ge3$.

What the mechanism can do. The folder index records that the induction cannot run inside $\omega_1^2$ with the target $\omega_1\omega$, because it needs the self-similar shape $\alpha\to(\alpha,3)^2$. The descent lemma (T4) removes that restriction at a price: one two-color relation per added color, each lowering the large target. Applied to the problem, if there are ordinals

$$\omega_1^2=\gamma_0\ge\gamma_1\ge\cdots\ge\gamma_k\ge\omega_1\omega$$

with $\gamma_{j-1}\to(\gamma_j,3)^2$ a theorem of ZFC for each $j$, then $\omega_1^2\to(\omega_1\omega,3,\ldots,3)^2_{k+1}$ is a theorem of ZFC, by (T4) and (T2). The first link is available: $\omega_1^2\to(\omega_1\alpha,3)^2$ for every $\alpha<\omega_1$, which Komjáth 2025 (Problem 13 commentary) attributes to a 1970 paper of Erdős and Hajnal; the problem page does not find it in that paper, and its case $\alpha=\omega$ is contained in the relation of Baumgartner and Hajnal. So $\gamma_1=\omega_1\alpha$ may be taken for any $\omega\le\alpha<\omega_1$. With $k=1$ this recovers the ZFC case $k=1$. For $k=2$ the route needs one further theorem of ZFC of the form $\gamma_1\to(\gamma_2,3)^2$ with $\omega_1\omega\le\gamma_2\le\gamma_1\le\omega_1^2$.

The obstruction. No chain of length two has both links theorems of ZFC, because under CH one of them fails:

• (a) Under CH, $\omega_1^2\not\to(\omega_1^2,3)^2$ (Hajnal 1971, [Ha71] on the page of Problem 1169 and on its erdosproblems.com page; the paper is not held). This excludes $\gamma_1=\omega_1^2$, since then the first link would be $\omega_1^2\to(\omega_1^2,3)^2$. A source qualification: Komjáth's Problem 13 commentary states Hajnal's hypothesis as $\kappa^{<\kappa}=\kappa$ for the relation $(\kappa^+)^2\not\to((\kappa^+)^2,3)^2$, which at $\kappa=\omega$ holds in ZFC and would contradict a later sentence of the same commentary, that the consistency of $\omega_1^2\to(\omega_1^2,3)^2$ is open; the CH form, which Bloom's catalog and the booklet item [Va99, 7.85] give, is used here.
• (b) Under CH, $\gamma\not\to(\omega_1\omega,3)^2$ for every ordinal $\gamma<\omega_1^2$ (author-recorded here). This excludes every $\gamma_1<\omega_1^2$, since $\gamma_1\to(\gamma_2,3)^2$ with $\gamma_2\ge\omega_1\omega$ gives $\gamma_1\to(\omega_1\omega,3)^2$ by (T2).

Proof of (b). Assume CH and let $W$ be a triangle-free graph on $\omega_1\omega$ with no independent set of order type $\omega_1\omega$: a witness for $\omega_1\omega\not\to(\omega_1\omega,3)^2$, which CH supplies by the Erdős--Hajnal theorem cited on Baumgartner's result page. Let $\alpha$ be a countable ordinal and fix an injection $f:\alpha\to\omega$. Write the elements of $\omega_1\alpha$ as $\omega_1\eta+\zeta$ with $\eta<\alpha$ and $\zeta<\omega_1$, calling $\eta$ the block index, and those of $\omega_1\omega$ as $\omega_1n+\zeta$. Put $\varphi(\omega_1\eta+\zeta)=\omega_1f(\eta)+\zeta$; $\varphi$ is injective and carries the block $\eta$ into the block $f(\eta)$ preserving order. Let $G$ be the graph on $\omega_1\alpha$ with $\{x,y\}\in G$ if and only if $\{\varphi(x),\varphi(y)\}\in W$. A triangle of $G$ maps to a triangle of $W$, so $G$ is triangle-free. Let $X\subseteq\omega_1\alpha$ be independent in $G$ with $\operatorname{otp}(X)=\omega_1\omega$, and let $X_\eta$ be its intersection with the block $\eta$. Then $\operatorname{otp}(X)$ is the sum, in the order of $\alpha$, of the $\operatorname{otp}(X_\eta)$, each at most $\omega_1$. If only $m$ of the $X_\eta$ were uncountable, the sum would be $\omega_1m+\rho$ for a countable $\rho$, because a sum of countably many countable ordinals is countable and $\rho'+\omega_1=\omega_1$ for countable $\rho'$; that is below $\omega_1(m+1)\le\omega_1\omega$. So infinitely many $X_\eta$ are uncountable, and $\varphi[X]$ meets infinitely many distinct blocks of $\omega_1\omega$ in uncountable sets, since $f$ is injective. The order type of $\varphi[X]$ is the sum over $n<\omega$ of the types of its intersections with the blocks; infinitely many terms equal $\omega_1$ (an uncountable subset of a block of type $\omega_1$ has type $\omega_1$), the rest are countable, and every term is at most $\omega_1$, so the sum is exactly $\omega_1\omega$. Thus $\varphi[X]$ is an independent set of $W$ of type $\omega_1\omega$, a contradiction. Hence $\omega_1\alpha\not\to(\omega_1\omega,3)^2$. For $\gamma<\omega_1^2$ write $\gamma=\omega_1\alpha_0+\zeta$ with $\alpha_0,\zeta<\omega_1$; then $\gamma<\omega_1(\alpha_0+1)$, and the restriction of $G$ for $\alpha=\alpha_0+1$ to the initial segment $\gamma$ is a witness for $\gamma\not\to(\omega_1\omega,3)^2$. $\square$

Conclusion. A chain of length $k\ge2$ needs $\gamma_1$ with both $\omega_1^2\to(\gamma_1,3)^2$ and $\gamma_1\to(\gamma_2,3)^2$, $\gamma_2\ge\omega_1\omega$; (a) refutes the first for $\gamma_1=\omega_1^2$ and (b) the second for $\gamma_1<\omega_1^2$, in every model of CH. So (T4) with triangle targets yields, from two-color relations that are theorems of ZFC, exactly the case $k=1$. In particular the ZFC case $k=2$ of Baumgartner and Hajnal cannot be obtained by (T4) from two-color relations $\gamma\to(\gamma',3)^2$ between ordinals in $[\omega_1\omega,\omega_1^2]$ that are theorems of ZFC, and (T4) cannot reduce the open case $k=3$ to such theorems. This sharpens the folder index's remark: not only the constant chain through $\omega_1\omega$ but every descending chain of length at least two has a link that fails under CH, so none consists of theorems of ZFC.

By-products. Statement (b) extends the two CH failures the review of Baumgartner's chapter cites, $\omega_1\omega\not\to(\omega_1\omega,3)^2$ and $\omega_1\omega^2\not\to(\omega_1\omega^2,3)^2$: under CH, $\gamma\not\to(\gamma,3)^2$ for every $\gamma$ with $\omega_1\omega\le\gamma<\omega_1^2$, by (T2). With Baumgartner's theorem, which under $\mathrm{MA}_{\aleph_1}$ gives $\gamma\to(\omega_1\omega,3)^2$ for every $\gamma\ge\omega_1\omega$ by (T2), the relation $\gamma\to(\omega_1\omega,3)^2$ is independent of ZFC for every $\gamma$ with $\omega_1\omega\le\gamma<\omega_1^2$, and the least $\gamma$ with the relation is $\omega_1\omega$ under $\mathrm{MA}_{\aleph_1}$ and $\omega_1^2$ under CH. A second by-product of the theorem page: Step 1 uses $\mathrm{MA}_{\aleph_1}$ only through Theorem A, so $\omega_1\omega\to(\omega_1\omega,3)^2$ alone implies the relation of Problem 1171 for every $k$. Hence a model of $\omega_1^2\not\to(\omega_1\omega,3,3,3)^2$, if there is one, is a model of $\omega_1\omega\not\to(\omega_1\omega,3)^2$, and the search for a consistent negative answer to the case $k=3$ can be confined to such models, for instance models of CH.

Next investigation. The mechanism is exhausted for the ZFC question. What remains useful is (i) to acquire the paper of Baumgartner and Hajnal (1987, not held) and identify the ingredient of the case $k=2$ that is not a two-color descent, since any attack on $k=3$ starts from that ingredient; and (ii) on the negative side, to look for a $4$-coloring of $[\omega_1^2]^2$ in a model of CH with no color-$0$ set of type $\omega_1\omega$ and no triangle in the other colors, CH being the simplest hypothesis under which such a witness can live. A positive outcome of (ii) would settle the case $k=3$ consistently in the negative; a proof that no such coloring exists under CH would leave the ZFC question open but exclude the natural models.

Outcome. Fails, with the obstruction (a)+(b); by-products recorded.

Problem 1169: two-color and finite-color negative relations on omega_1^2

Target. Problem 1169 asks whether $\omega_1^2\not\to(\omega_1^2,3)^2$. The page carries the status open and a compiled assessment: ZFC does not refute the relation, and whether ZFC proves it is open. The erdosproblems.com page (accessed 2026-09-28; last edited 25 January 2026) states the problem with a quantifier "for all finite $k<\omega$" that the displayed relation does not use, records that Hajnal proved the negative relation from CH [Ha71], and lists no proof claim. The booklet item [Va99, 7.85], read in the transcription of the booklet, has no quantifier: "Is it true that $\omega_1^2\not\to(\omega_1^2,3)^2$? [Hajnal show[ed] this follows from CH.]" (the verb is cut at the scan's right edge). The open part is whether the negative relation is a theorem of ZFC, equivalently whether $\omega_1^2\to(\omega_1^2,3)^2$ is consistent, which Komjáth 2025 (Problem 13 commentary) calls open as far as the author knows.

What transfers. (T5) at $\alpha=\omega_1^2$: for every finite $k\ge1$,

$$\omega_1^2\not\to(\omega_1^2,3)^2 \iff \omega_1^2\not\to(\omega_1^2,\underbrace{3,\ldots,3}_{k})^2_{k+1}.$$

So the stray quantifier is harmless: read with $k$ triangle colors, the erdosproblems.com statement is equivalent to the booklet's for every $k$, and a witness for the many-color relation contains a two-color witness on a subset of type $\omega_1^2$. Second, by (T3) and (T2), every model of $\omega_1^2\to(\omega_1^2,3)^2$ satisfies $\omega_1^2\to(\omega_1^2,3,\ldots,3)^2_{k+1}$ for every $k$, hence the relation of Problem 1171 with the target $\omega_1^2$ in place of $\omega_1\omega$. This is the folder index's remark, now with its proof. The conclusion is stronger than what $\mathrm{MA}_{\aleph_1}$ gives for Problem 1171, but its premise is Komjáth's open consistency question, and nothing here attacks that premise.

What does not transfer. Nothing bears on the ZFC question. The mechanism produces positive relations from positive relations; toward the negative relation it says only, through (T5), that adding colors does not make a witness easier to build. Context, not used: an arXiv preprint (Golshani, arXiv:2608.13213, August 2026; unrefereed, not held; its abstract was read) states that a finite-support iteration of $\sigma$-centered forcings over a model of CH carrying Hajnal's coloring yields a model of $\mathrm{MA}_{\omega_1}(\sigma\text{-centered})$, $2^{\aleph_0}=\aleph_2$ and $\omega_1^2\not\to(\omega_1^2,3)^2$, and that under $\mathrm{MA}_{\omega_1}$ every c.c.c. forcing preserves an existing witness. If so, the negative relation is compatible with the fragment $\mathrm{MA}_{\omega_1}(\sigma\text{-centered})$, whereas $\omega_1\omega\not\to(\omega_1\omega,3)^2$ contradicts full $\mathrm{MA}_{\aleph_1}$ (Theorem A on the theorem page); whether that fragment alone excludes $\omega_1\omega\not\to(\omega_1\omega,3)^2$ is not settled by anything held here; and the iteration the abstract describes does not reach a model of the positive relation. The search log on the Problem 1171 page lists the preprint.

Next investigation and outcome. None from this mechanism. Outcome: transfers conditionally, an equivalence and a conditional strengthening; no lead, since the premise is the open problem itself and the unconditional conclusion for Problem 1171 is already known by the $\mathrm{MA}_{\aleph_1}$ route.

Problem 597: graph targets

Target. Problem 597 asks two things: (Q1) for every graph $G$ on at most $\aleph_1$ vertices with no $K_4$ and no $K_{\aleph_0,\aleph_0}$, is $\omega_1^2\to(\omega_1\omega,G)^2$; and (Q2) what about finite $G$. The page carries the status open and a compiled account of its claim pages. The erdosproblems.com page (accessed 2026-09-28; last edited 23 January 2026) records that Erdős and Hajnal proved $\omega_1^2\to(\omega_1\omega,3)^2$, that Erdős first asked the question for $K_4$-free $G$ only, and that Baumgartner proved $\omega_1^2\not\to(\omega_1\omega,K_{\aleph_0,\aleph_0})^2$ (no hypothesis is stated on the page), which is why $K_{\aleph_0,\aleph_0}$ is excluded; the exclusion of $K_4$ is forced by $\omega_1^2\not\to(\omega_1\omega,4)^2$ under CH (the negative relation).

(Q2) under Martin's axiom: transfers, without the lemma. Under $\mathrm{MA}_{\aleph_1}$, Baumgartner's theorem at full strength gives $\omega_1\omega\to(\omega_1\omega,n)^2$ for every finite $n$ (main theorem; the statement is taken from the zbMATH review and one held restatement (an arXiv preprint), the chapter is not held, and the deposit cites only the case $n=3$). For a finite graph $G$ on $n$ vertices, $G$ embeds in $K_n$, so $\omega_1\omega\to(\omega_1\omega,G)^2$ by (T2), and restriction to the initial segment, Steps 2 and 3 of the theorem page, gives $\omega_1^2\to(\omega_1\omega,G)^2$. Hence under $\mathrm{MA}_{\aleph_1}$ the answer to (Q2) is yes for every finite $G$, whether or not $G$ is $K_4$-free; with the relative consistency of $\mathrm{MA}_{\aleph_1}$ (Theorem C on the theorem page), (Q2) is not disprovable in ZFC. This is a deduction from the imported theorem and the restriction step alone; the color-merging induction is not needed. What the induction adds are the finite-color variants: by (T3) at $\alpha=\omega_1\omega$, under $\mathrm{MA}_{\aleph_1}$, $\omega_1^2\to(\omega_1\omega,G_1,\ldots,G_k)^2_{k+1}$ for any finite graphs $G_i$, which also follows from Baumgartner's theorem with $n$ the multicolor Ramsey number of the cliques on the $G_i$; and, under any hypothesis giving $\omega_1\omega\to(\omega_1\omega,G_i)^2$ for infinite $G_i$, the same relation with those $G_i$, which the Ramsey bridge cannot give.

(Q2) in ZFC: the mechanism does not apply, but an implication does. (Q2) is a two-color relation whose large target $\omega_1\omega$ lies below the resource $\omega_1^2$. The induction adds colors and cannot enlarge a triangle target to $G$, and (T4) would need a two-color relation with the target $G$ on some ordinal at most $\omega_1^2$, which is (Q2) itself or stronger. So the mechanism is orthogonal to the question. In the other direction (author-recorded): for each finite $k$ let $F_k$ be a finite $K_4$-free graph such that every $k$-coloring of its edges has a monochromatic triangle, which exists for $k=2$ by Folkman's Theorem 1 and for every $k$ by the theorem of Nešetřil and Rödl recorded on Problem 924. 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: given a coloring with the colors $0,\ldots,k$, merge the colors $1,\ldots,k$ into one; a color-$0$ set of type $\omega_1\omega$ is what is wanted, and a copy of $F_k$ in the merged color carries a $k$-coloring of its edges, hence a monochromatic triangle. So a positive answer to (Q2) in ZFC for the graphs $F_k$ would answer Problem 1171 in ZFC for every $k$, and every model of $\omega_1^2\not\to(\omega_1\omega,3,\ldots,3)^2_{k+1}$ is a model in which (Q2) fails for the $K_4$-free graph $F_k$. (Q2) for Folkman-type graphs is therefore at least as strong as the open case $k=3$ of Problem 1171. The converse implication gives nothing: the relation of Problem 1171 yields (Q2) only for subgraphs of a triangle.

(Q1): not reached. The route under $\mathrm{MA}_{\aleph_1}$ would need $\omega_1\omega\to(\omega_1\omega,G)^2$ for infinite $G$ with the stated exclusions; Baumgartner's theorem is stated for finite $n$ only, and nothing held decides the infinite case. A necessary condition for (Q1) in a model is that every graph on $\omega_1^2$ without an independent set of type $\omega_1\omega$ contains, as a subgraph, every such $G$; the $K_{\aleph_0,\aleph_0}$ exclusion is exactly the failure of that condition at $K_{\aleph_0,\aleph_0}$.

Next investigation. (i) Acquire Baumgartner's chapter and check the full-strength statement on its page images; until then the answer to (Q2) under $\mathrm{MA}_{\aleph_1}$ rests on the review and the restatement. (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 positive outcome makes (Q1) not disprovable for the $G$ covered; 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, only if some model satisfies $\omega_1^2\not\to(\omega_1\omega,G)^2$ for a $G$ with the exclusions of (Q1), which a failure on $\omega_1\omega$ does not give: under CH, $\omega_1\omega\not\to(\omega_1\omega,3)^2$ while $\omega_1^2\to(\omega_1\omega,3)^2$ is a theorem of ZFC. (iii) For (Q2) in ZFC, test the smallest Folkman-type graphs first, knowing that a positive answer for $F_3$ settles the case $k=3$ of Problem 1171 and that a consistent negative answer for that case refutes (Q2) at $F_3$.

Outcome. Transfers conditionally for (Q2) under $\mathrm{MA}_{\aleph_1}$, from the imported theorem and the restriction step; fails for the ZFC question and for (Q1), with the implication to Problem 1171 recorded.

Problems 590, 591, 592 and 118: countable partition ordinals

Targets. Problem 590 (status solved, claim proved; the erdosproblems.com label PROVED (LEAN)): $\omega^\omega\to(\omega^\omega,3)^2$, Chang's theorem (theorem_p396), with $\omega^\omega\to(\omega^\omega,m)^2$ for every finite $m$ reported from Milner and Larson on the same page. Problem 591 (status solved, claim proved): $\omega^{\omega^2}\to(\omega^{\omega^2},3)^2$, the case $\beta=2$ of Schipperus's Theorem 28. Problem 592 (status open): determine the countable $\beta$ with $\omega^{\omega^\beta}\to(\omega^{\omega^\beta},3)^2$; known are yes for $\beta$ a sum of one or two indecomposables (Theorem 28) and no for four or more (Theorem 29, part 3), with three undecided. Problem 118 (status solved, claim disproved): whether $\alpha\to(\alpha,3)^2$ forces $\alpha\to(\alpha,n)^2$ for every $n$, refuted by $\omega^{\omega^2}\not\to(\omega^{\omega^2},6)^2$ (Theorem 29, part 1) and, per Schipperus's report, by Larson's $\not\to(\omega^{\omega^2},5)^2$ together with Larson's $\omega^{\omega^2}\to(\omega^{\omega^2},4)^2$ (not held). Each of these pages carries a compiled assessment.

What transfers. Lemma 2.1 applies verbatim to every countable partition ordinal: if $\alpha\to(\alpha,3)^2$ then $\alpha\to(\alpha,3,\ldots,3)^2_{k+1}$ for every finite $k$, and by (T5) conversely. Three consequences:

• Problem 592 is unchanged by adding triangle colors. For every countable $\beta$ and every $k\ge1$, $\omega^{\omega^\beta}\to(\omega^{\omega^\beta},3)^2$ if and only if $\omega^{\omega^\beta}\to(\omega^{\omega^\beta},3,\ldots,3)^2_{k+1}$. This is an equivalent reformulation, and it enables nothing new: the extracted witness of (T5) lives on a subset of full type and needs no new construction, and a positive many-color relation is no easier to prove than the two-color one it contains. The undecided case is treated in the next section.
• Problem 591's relation strengthens in the color direction where it stops in the clique direction. By Theorem 28, $\omega^{\omega^2}\to(\omega^{\omega^2},3,\ldots,3)^2_{k+1}$ for every $k$, although $\omega^{\omega^2}\not\to(\omega^{\omega^2},6)^2$. Here the deposit's bridge is strictly stronger than the finite-Ramsey bridge of Baumgartner's result page: already the case $k=2$, $\omega^{\omega^2}\to(\omega^{\omega^2},3,3)^2_3$, would need $\omega^{\omega^2}\to(\omega^{\omega^2},6)^2$ through the Ramsey number $R(3,3)=6$, which is false. For Problem 590 the two bridges agree, since $\omega^\omega\to(\omega^\omega,m)^2$ holds for every finite $m$.
• Problem 118's color sibling has a positive answer. The disproved question asks whether $\alpha\to(\alpha,3)^2$ forces $\alpha\to(\alpha,n)^2$; its sibling, whether $\alpha\to(\alpha,3)^2$ forces $\alpha\to(\alpha,3,\ldots,3)^2_{k+1}$, holds for every cardinal, ordinal or order type $\alpha$ and every finite $k$, by Lemma 2.1 with (T1). Raising the clique target is not free; adding triangle colors is.

A ZFC instance and its finite corollary. At $\alpha=\omega^{\omega^2}$, Theorems 28 and 29(1) with Lemma 2.1 give, in ZFC and from held sources,

$$\alpha\not\to(\alpha,6)^2 \quad\text{and}\quad \alpha\to(\alpha,3,3,3,3,3)^2_6,$$

and with Larson's sharpening (not held) the same with $5$ and four triangle colors. This is a cousin of the question Komjáth records (next section), which asks for $4$ against three triangle colors. It has a finite corollary in the pattern of Hajnal's forcing-and-compactness argument (Komjáth 2025, Problem 54 discussion), without the forcing: let $X$ be the color-$1$ graph of a witness for $\alpha\not\to(\alpha,6)^2$, so $X$ is $K_6$-free with no independent set of type $\alpha$. Every $5$-coloring of the edges of $X$ has a monochromatic triangle: extend it by the color $0$ on the non-edges; a color-$0$ set of type $\alpha$ would be independent in $X$, so some color $1,\ldots,5$ has a triangle. By compactness some finite subgraph of $X$ has the same property. The finite statement is weaker than the theorem of Nešetřil and Rödl on Problem 924, which gives $K_4$-free graphs for every number of colors; only the derivation through a countable ordinal is new here.

Outcome. Transfers, as an equivalence for Problem 592 and as corollaries for Problems 590, 591 and 118; no route to the undecided case of Problem 592 and no lead from this section.

Problems 592 and 1171: four-clique versus triangle colors

Target. Komjáth 2025 (Problem 54 discussion) records, next to the unknown case $k=3$ of Problem 1171, that it is unknown "at least" whether it is consistent that $\alpha\not\to(\beta,4)^2$ and $\alpha\to(\beta,3,3,3)^2$ hold together for some ordinals $\alpha,\beta$; the Problem 1171 page quotes this in its assessment. The undecided case of Problem 592 is $\omega^{\omega^\beta}\to(\omega^{\omega^\beta},3)^2$ for $\beta$ a sum of exactly three indecomposable ordinals, the smallest being $\beta=3$ and $\alpha=\omega^{\omega^3}$; the outside status was checked against the report on the Isabelle formalization of ordinal partition relations by Džamonja, Koutsoukou-Argyraki and Paulson (arXiv:2011.13218, 2020; not held in the library, its text layer read from a downloaded copy), which states Schipperus's theorem as its Theorem 2.2 and then says that the case of at least three summands in the Cantor normal form is still open; a web search found no later result.

The deduction (author-recorded). If an ordinal $\alpha$ satisfies $\alpha\to(\alpha,3)^2$ and $\alpha\not\to(\alpha,4)^2$, then by Lemma 2.1 $\alpha\to(\alpha,3,3,3)^2_4$, so the pair $(\alpha,\alpha)$ answers Komjáth's question, in ZFC rather than merely consistently. Theorem 29(2) supplies the second half for every $\alpha=\omega^{\omega^\beta}$ with countable $\beta$ a sum of three indecomposables (read on the page images at statement depth; the proofs of Theorems 31--33 were read and not checked); the first half is exactly the undecided case of Problem 592. So: if $\omega^{\omega^\beta}\to(\omega^{\omega^\beta},3)^2$ for some countable $\beta$ that is a sum of three indecomposables, then Komjáth's question has a positive answer in ZFC. The pair with $\alpha=\omega^{\omega^2}$ does not serve, since $\omega^{\omega^2}\to(\omega^{\omega^2},4)^2$ by Larson's theorem; and any answer $(\alpha,\beta)$ has $\beta\not\to(\beta,4)^2$, because $\beta\to(\beta,4)^2$ would give $\alpha\to(\beta,4)^2$ by (T2). The other shape of an answer, with $\beta<\alpha$, contains the case $k=3$ of Problem 1171 under CH: $(\omega_1^2,\omega_1\omega)$ is an answer in a model of CH exactly when $\omega_1^2\to(\omega_1\omega,3,3,3)^2$ holds there, by the negative relation of Baumgartner and Hajnal.

Falsification checks. The quantifiers match: Komjáth asks for some $\alpha,\beta$ and some model; the deduction gives one pair in every model. The targets match: Theorem 29's $\not\to(\cdot,4)^2$ is the two-color negative relation with a four-element homogeneous set, the same convention as the lemma page. Nothing here decides the premise.

Next investigation. Settle $\omega^{\omega^\beta}\to(\omega^{\omega^\beta},3)^2$ for $\beta$ a sum of three indecomposables, starting at $\beta=3$. Schipperus's proof of Theorem 28 says that only its triangle-in-color-$1$ step, Lemma 26, is sensitive to the number of indecomposables, so the first task is to see whether that step, or a replacement, survives a third summand. A positive outcome answers Komjáth's question in ZFC and extends the positive boundary of Problem 592; a negative outcome for every such $\beta$ completes the classification of Problem 592 as "at most two indecomposables" ($\beta=0$ included, since $\omega\to(\omega,3)^2$) and leaves Komjáth's question to the other shape of answer, or to the shape $(\alpha,\alpha)$ at a partition ordinal whose four-clique relation is undecided, such as $\omega^{\omega^\omega}$, where Theorem 28 gives the $3$-relation and nothing held decides the $4$-relation.

Outcome. Transfers conditionally; filed as the lead four-clique versus triangle colors.

Problems 1174, 595 and 596: countably many colors

Targets. Problem 1174 (status open, each part with one consistency side settled): a $K_4$-free graph every countable edge coloring of which has a monochromatic triangle, and a $K_{\aleph_1}$-free graph every countable edge coloring of which has a monochromatic $K_{\aleph_0}$. Problem 595 (status open): a $K_4$-free graph that is not a union of countably many triangle-free graphs, which is the first question of Problem 1174 in other words, since a countable union of triangle-free graphs is a countable edge coloring without monochromatic triangles. Komjáth 2025 (Problem 53) records Shelah's consistency proof (Shelah 1989, Lemma 5.1, relative to a measurable cardinal or the weaker partition hypothesis the lemma allows in its place) and that existence in ZFC is open. Problem 596 (status open): the pairs $(G_1,G_2)$ for which $G_1$-free graphs force a monochromatic $G_2$ under every finite number of colors but not under $\aleph_0$ colors; the pair $(K_4,K_3)$ is Problem 595.

Why the mechanism fails. The induction adds one color per step and has no limit step: the case $k$ is used once to reach a set of full type two-colored by $\{0,1\}$, and for $\aleph_0$ colors the corresponding merging leaves infinitely many colors on that set, with nothing to apply. The obstruction is not a defect of the proof but of the statement: for every order type $\alpha$ with $2\le|\alpha|\le2^{\aleph_0}$,

$$\alpha\not\to(\alpha,\underbrace{3,3,\ldots}_{\aleph_0})^2_{\aleph_0}$$

(author-recorded; the fact is the one Komjáth states for Problems 51 and 52, that $K_{2^{\aleph_0}}$ is a union of $\aleph_0$ bipartite graphs). Fix an injection $\iota:\alpha\to2^\omega$ and color $\{x,y\}$ by $1+n$, where $n$ is the least position at which $\iota(x)$ and $\iota(y)$ differ; leave the color $0$ unused. A triangle homogeneous in color $1+n$ would need three pairwise distinct values at position $n$, and there are two. So on $\omega_1^2$, on $\omega_1\omega$ and on every countable ordinal the countable-color relation with triangle targets fails outright, and no ordinal partition relation on these ordinals can supply the graph of Problem 1174 or 595, which must have more than $2^{\aleph_0}$ vertices; Shelah's consistent example has $2^\lambda$ vertices for a suitable $\lambda$. The same coloring, with the color $0$ unused, refutes the countable-color form of every relation on this page whose ordinal has at most $2^{\aleph_0}$ elements and whose targets in the colors $\ge1$ each contain an odd cycle (a triangle, a larger clique, an ordinal $\ge3$, or a graph containing a triangle): its color classes are bipartite, so it says nothing about the bipartite targets of Problems 597 and 601, and nothing above the continuum, where $(2^{\aleph_0})^+\to((2^{\aleph_0})^+,(\aleph_1)_{\aleph_0})^2$ holds (Erdős and Rado, displayed by Komjáth 2025 under Problem 53). For Problem 596 the mechanism says only that the two-color finite side of the pair $(K_4,K_3)$ is reachable from ordinal relations, by Hajnal's argument; that for three or more colors the same route needs $\alpha\not\to(\beta,4)^2$ and $\alpha\to(\beta,3,3,3)^2$ in one model, which Komjáth 2025 (p. 443) records as unknown (see the section on Problems 582 and 924); and that the countable side is out of reach, by the cardinality obstruction.

Outcome. Fails, with the cardinality obstruction; no lead.

Problem 1168

Target. Problem 1168 (status open): prove $\aleph_{\omega+1}\not\to(\aleph_{\omega+1},3,\ldots,3)^2_{\aleph_0}$ without GCH. The library card of Garti, Hayut and Shelah records the classical case $2^\lambda=\lambda^+$ and the consistency of the negative relation with $2^\lambda>\lambda^+$ under pcf and guessing hypotheses.

What transfers. Only the finite-color positives, which are trivial: by the Erdős--Dushnik--Miller theorem $\kappa\to(\kappa,\omega)^2$ for every infinite cardinal $\kappa$, so $\kappa\to(\kappa,3)^2$ and, by Lemma 2.1, $\kappa\to(\kappa,3,\ldots,3)^2_{k+1}$ for every finite $k$; the Ramsey bridge gives the same from $\kappa\to(\kappa,n)^2$. The problem asks for the failure at $\aleph_0$ colors, where the induction has no limit step. One remark of the previous section applies: if $\aleph_{\omega+1}\le2^{\aleph_0}$ the bit coloring refutes the relation at once, so the content of the problem is the case $2^{\aleph_0}<\aleph_{\omega+1}$, which is where the cited work operates.

Outcome. Fails; no lead.

Problem 601: the infinite-path target

Target. Problem 601 (status open): for which limit ordinals $\alpha$ does every graph on $\alpha$ have an infinite path or an independent set of order type $\alpha$; in the notation of (T1), $\alpha\to(\alpha,P)^2$ with $P$ the one-way infinite path as a subgraph.

What transfers. (T3) with $H_1=\cdots=H_k=P$: if $\alpha$ has the property, then for every finite $k$, whenever $X_1,\ldots,X_k$ are graphs on $\alpha$ none of which contains an infinite path, their union has an independent set of type $\alpha$. Proof: color a pair by the least $i$ with the pair in $X_i$, and by $0$ if there is none; a color-$i$ infinite path is a path of $X_i$, so (T3) gives a color-$0$ set of type $\alpha$, independent in every $X_i$. The union of two graphs without infinite paths can contain one, so this is not immediate from the definition, but it says nothing about which $\alpha$ have the property, which is the whole question; the $\mathrm{MA}$ side of the problem (Larson 1990, [La90] on the page, not held) is not in the corpus.

Outcome. Transfers, without bearing on the question; no lead.

Problems 582 and 924: finite Folkman graphs

Both are proved: Problem 582 by Folkman's Theorem 1 and Problem 924 by Nešetřil and Rödl. Hajnal's argument (Komjáth 2025, Problem 54 discussion) derives Folkman's two-color graph from the ZFC relation $\omega_1^2\to(\omega_1\omega,3,3)^2$, the CH witness for $\omega_1^2\not\to(\omega_1\omega,4)^2$, compactness, and the absoluteness of finite graphs. Its $k$-color analog through Problem 1171 would need the ZFC relation with $k$ triangle targets, open for $k\ge3$, and the CH witness in the same model; under $\mathrm{MA}_{\aleph_1}$, where the relation is known, no $K_4$-free witness exists, since Baumgartner's theorem gives $\omega_1^2\to(\omega_1\omega,4)^2$ there. So a positive answer to the open case of Problem 1171 would only re-derive, by the same argument, the triangle case of finite theorems already proved, and the finite theorems give nothing back to Problem 1171; the only derivation of a Folkman-type finite graph from the material here is the $K_6$-free, five-color instance at $\omega^{\omega^2}$ above, which the finite theorems exceed. Outcome: nothing to transfer in either direction; no lead.

Other relations in the set-theory folder

The mechanism needs a color-$0$ target equal to the ambient order type, or a descending chain as in (T4), fixed targets in the other colors, and finitely many colors; the exponent plays no part in its proofs. It does not apply to Problem 1170 and Problem 1172, whose relations, one for Problem 1170 and five under GCH for Problem 1172, all have both targets below the resource; nor to Problem 70, $\mathfrak{c}\to(\beta,n)^3_2$ for countable $\beta$ and finite $n$, whose color-$0$ target lies below the resource; nor to Problem 1219, a cardinal relation with both targets below the resource. For Problem 1218, $\lambda\not\to(\lambda,4,4,\ldots)^3$ with countably many colors under GCH, the mechanism runs in exponent $3$: for every finite $k$ the negative relation with $k$ colors carrying $4$ is equivalent to $\lambda\not\to(\lambda,4)^3$, which implies the asked relation by (T2); the induction does not reach countably many colors, and that, not the exponent, is the obstruction. For Problem 1220, $\lambda\to(\lambda,\aleph_1)^2$ for a singular cardinal $\lambda$ with $\lambda$ and $\mathrm{cf}(\lambda)$ both $\aleph_0$-inaccessible, (T3) would give $\lambda\to(\lambda,\aleph_1,\ldots,\aleph_1)^2_{k+1}$ from the relation asked. The problem page records the question as independent of ZFC: under GCH the relation holds for every such $\lambda$ (Erdős, Hajnal and Rado), and in a model of Shelah and Stanley it fails for one; so (T3) gives the $(k+1)$-color relation for every such $\lambda$ under GCH, a corollary that plays no part in settling the problem. The canonization mechanism of the Problem 1219 connections page handles Problem 1220's strong-limit case and records the obstruction beyond it. None of these is pursued.

Outcomes

• Problem 1171, ZFC case $k\ge3$: fails. The descent lemma (T4) reduces the case $k$ to a chain of $k$ two-color relations, and under CH no chain of length two exists, by Hajnal's theorem and statement (b); by-products are the CH failure $\gamma\not\to(\omega_1\omega,3)^2$ for all $\gamma<\omega_1^2$ and the reduction of a consistent negative answer to models of $\omega_1\omega\not\to(\omega_1\omega,3)^2$.
• Problem 1169: transfers conditionally. The two-color and $k$-color negative relations on $\omega_1^2$ are equivalent for every $k$, and a model of the positive relation would satisfy Problem 1171's relation with target $\omega_1^2$.
• Problem 597: transfers conditionally for finite $G$ under $\mathrm{MA}_{\aleph_1}$, from Baumgartner's theorem and the restriction step; fails for the ZFC question and for infinite $G$; (Q2) for Folkman-type graphs implies Problem 1171 in ZFC.
• Problems 590, 591, 592, 118: transfers as an equivalence (592) and as corollaries (590, 591, 118); the ZFC instance "$6$ against five triangle colors" at $\omega^{\omega^2}$ is recorded.
• Komjáth's question "$4$ against three triangle colors" (Problems 1171 and 592): transfers conditionally on the undecided three-indecomposable case of Problem 592. Lead filed.
• Problems 1174, 595, 596 and 1168: fail; countably many colors are out of reach of the induction, and the countable-color relation is false on every order type of size at most the continuum.
• Problem 601: transfers without bearing on the question.
• Problems 582, 924 and the relations of the last section: not applicable, Problem 1218 by the countable color count.