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Source. Lezhe Gao, A finite-color partition relation for $\omega_1^2$ under $\mathrm{MA}_{\aleph_1}$, Theorem 3.1, stated on physical p. 3 and proved on physical pp. 3--4 (§3, "The main theorem"; the physical and printed page numbers agree), in the four-page PDF held by its library source card, Gao (2026); the corpus files the result as Theorem 3.1. The relation the proof imports is displayed as relation (1) on p. 2, and the lemma it uses is reconstructed on the Lemma 2.1 page.

Standing. This is an author-recorded reconstruction; it is not an independent review, changes no status and assigns no tier. The deposit is unrefereed. The argument is conditional on $\mathrm{MA}_{\aleph_1}$ through one imported theorem whose proof is not held in the corpus (Theorem A below); the deductions the deposit itself makes are written out in full.

Definitions

The partition relation $\alpha\to(\beta_0,\ldots,\beta_{n-1})^2_n$, colorings, homogeneous sets, triangles and order types are as defined on the Lemma 2.1 page.

Ordinal arithmetic. $\omega_1$ is the first uncountable ordinal. $\omega_1\omega$ is the ordinal product $\omega_1\cdot\omega$, the order type of $\omega$ copies of $\omega_1$ laid end to end, equal to $\sup_{n<\omega}\omega_1\cdot n$; and $\omega_1^2=\omega_1\cdot\omega_1$. Ordinal multiplication by a fixed nonzero left factor is strictly increasing in the right factor, and $\omega<\omega_1$, so $\omega_1\omega<\omega_1^2$. With von Neumann ordinals this makes $\omega_1\omega$ a subset of $\omega_1^2$, namely the set of ordinals below $\omega_1\omega$, and an initial segment of it: every element of $\omega_1^2$ below an element of $\omega_1\omega$ belongs to $\omega_1\omega$. The order that $\omega_1\omega$ inherits as a subset of $\omega_1^2$ is membership, the same as its own order, so its order type as a subset of $\omega_1^2$ is $\omega_1\omega$.

Martin's axiom for $\aleph_1$ dense sets. $\mathrm{MA}_{\aleph_1}$ states that for every partial order with the countable chain condition and every family of at most $\aleph_1$ dense subsets of it there is a filter meeting every member of the family. It enters the argument only as the hypothesis of Theorem A; no forcing argument is made here. It implies $2^{\aleph_0}>\aleph_1$, so it contradicts the continuum hypothesis.

Imported theorems

Theorem A (Baumgartner 1989, §3, the case $n=3$). Assume $\mathrm{MA}_{\aleph_1}$. Then

$$\omega_1\omega\to(\omega_1\omega,3)^2.$$

Exact version and provenance. The chapter Baumgartner (1989) is not held (paywalled). Its zbMATH review, Zbl 0703.03027, reports that §3 proves that $\mathrm{MA}_{\aleph_1}$ makes $\omega_1\omega$ and $\omega_1\omega^2$ partition ordinals, that is, $\alpha\to(\alpha,n)^2$ for every finite $n$, and the held paper of Chen, Garti and Weinert restates the $\omega_1\omega$ half in that form; the corpus files it as the main theorem. Gao's deposit cites the chapter for the case $n=3$ only, as its relation (1) on p. 2, and that case is all the proof below uses. Theorem A is consumed as an external input; its proof is not reconstructed, since its text is not held, and the whole argument is conditional on it.

Theorem B (Baumgartner and Hajnal 1987; not used in the proof). In ZFC,

$$\omega_1^2\to(\omega_1\omega,3,3)^2,$$

the case $\kappa=\omega$ of $(\kappa^+)^2\to(\kappa^+\kappa,3,3)^2$ for regular $\kappa$ with $\kappa^{<\kappa}=\kappa$. The deposit's introduction (p. 2) names it as the case $k=2$ of the problem, already known in ZFC. It is not a premise of Theorem 3.1 and is recorded here only to fix the exact version behind that remark. The paper is not held; the statement is taken from its zbMATH review, Zbl 0635.03042, and from Komjáth 2025, and is filed as the positive relation.

Theorem C (Solovay and Tennenbaum 1971; the step from the theorem to the catalog status). If ZFC is consistent, then so is ZFC together with Martin's axiom and $2^{\aleph_0}>\aleph_1$, and hence ZFC together with $\mathrm{MA}_{\aleph_1}$. The deposit does not state this step; the problem page uses it to pass from Theorem 3.1 to "not disprovable in ZFC". The paper is not held and is cited on the problem page as [SoTe71].

Statement

Assume $\mathrm{MA}_{\aleph_1}$. Then for every finite $k\ge1$,

$$\omega_1^2\to(\omega_1\omega,\underbrace{3,\ldots,3}_{k})^2_{k+1}:$$

every coloring of $[\omega_1^2]^2$ with the colors $0,\ldots,k$ has a subset of $\omega_1^2$ of order type $\omega_1\omega$ homogeneous in color $0$, or a triangle of some color in $\{1,\ldots,k\}$.

Proof

Assume $\mathrm{MA}_{\aleph_1}$ and fix a finite $k\ge1$.

Step 1: the relation on $\omega_1\omega$. By Theorem A the ordinal $\alpha=\omega_1\omega$ satisfies $\alpha\to(\alpha,3)^2$. Lemma 2.1 with this $\alpha$ gives

$$\omega_1\omega\to(\omega_1\omega,\underbrace{3,\ldots,3}_{k})^2_{k+1}.$$

Step 2: restriction to the initial segment. Let $c:[\omega_1^2]^2\to\{0,\ldots,k\}$ be any coloring. Since $\omega_1\omega\subseteq\omega_1^2$, every two-element subset of $\omega_1\omega$ is a two-element subset of $\omega_1^2$, so $c_0=c\upharpoonright[\omega_1\omega]^2$ is a coloring of $[\omega_1\omega]^2$ with $k+1$ colors. Step 1 applied to $c_0$ gives either a set $X\subseteq\omega_1\omega$ with $\operatorname{otp}(X)=\omega_1\omega$ and $c_0$ constantly $0$ on $[X]^2$, or a three-element set $T\subseteq\omega_1\omega$ on whose pairs $c_0$ is constant with a value in $\{1,\ldots,k\}$.

Step 3: reading the result in $\omega_1^2$. The sets $X$ and $T$ are subsets of $\omega_1^2$. The order they inherit from $\omega_1^2$ is membership, the same order they inherit from $\omega_1\omega$, so $\operatorname{otp}(X)=\omega_1\omega$ as a subset of $\omega_1^2$, and $c$ agrees with $c_0$ on $[X]^2$ and on $[T]^2$. Hence under $c$ either $X$ is a subset of $\omega_1^2$ of order type $\omega_1\omega$ homogeneous in color $0$, or $T$ is a triangle of some color in $\{1,\ldots,k\}$. The coloring $c$ was arbitrary, so the relation holds. This proves the theorem.

Step 2 uses only that $\omega_1\omega$ is a subset of $\omega_1^2$ carrying the inherited order; that it is an initial segment is more than is needed. In general, if $\beta\to(\beta_0,\ldots,\beta_{n-1})^2_n$ and $\alpha$ has a subset of order type $\beta$, then $\alpha\to(\beta_0,\ldots,\beta_{n-1})^2_n$, by the same restriction and the transport fact of the lemma page.

Fidelity and scope

• The catalog question. The catalog asks the relation for all finite $k<\omega$ with $k$ triangle targets and $k+1$ colors. The theorem covers every $k\ge1$. The instance $k=0$ is the one-color relation $\omega_1^2\to(\omega_1\omega)^2_1$, which holds outright: under the only coloring with one color the subset $\omega_1\omega$ of $\omega_1^2$ is homogeneous of order type $\omega_1\omega$. So, under $\mathrm{MA}_{\aleph_1}$, every instance of the catalog question has a positive answer.
• What is proved and what is not. The relation is proved from $\mathrm{MA}_{\aleph_1}$; with Theorem C it is not disprovable in ZFC. It is not proved in ZFC, and Komjáth 2025 records the ZFC instance $k=3$ as unknown. The route cannot be run in ZFC: the continuum hypothesis gives $\omega_1\omega\not\to(\omega_1\omega,3)^2$ (Erdős and Hajnal, as the review of Baumgartner 1989 reports), so Step 1 needs a hypothesis beyond ZFC even though the conclusion holds in ZFC for $k\le2$ by Theorem B.
• A source qualification. The introduction (p. 2) says that the case $k=1$ "is exactly" relation (1). Read against Theorem 3.1, the case $k=1$ is $\omega_1^2\to(\omega_1\omega,3)^2$, a theorem of ZFC that Komjáth 2025 attributes to Erdős and Hajnal (1970), whereas relation (1) is $\omega_1\omega\to(\omega_1\omega,3)^2$, the case $k=1$ of the intermediate relation of Step 1. The slip is confined to the introduction and does not affect the proof.

Depends on. Theorem A, imported with its proof not held, and Lemma 2.1; Theorem C is used only for the passage from the theorem to the catalog status.

Bears on. Problem 1171, as a conditional proof under $\mathrm{MA}_{\aleph_1}$ whose status the page already records.