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Source. Lezhe Gao, A finite-color partition relation for $\omega_1^2$ under $\mathrm{MA}_{\aleph_1}$, Lemma 2.1, stated on physical p. 2 and proved on physical pp. 2--3 (§2, "A general color-reduction lemma"; 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 Lemma 2.1. Remark 3.2 on p. 4 restates the conclusion as the stability of $\alpha\to(\alpha,3)^2$ under adding finitely many colors with target $3$. The source calls the lemma standard and includes the proof for completeness.

Standing. This is an author-recorded reconstruction of the deposit's argument; it is not an independent review, changes no status and assigns no tier. The deposit is unrefereed. The lemma uses nothing beyond its hypothesis: no axiom beyond ZFC enters, and no external theorem is imported.

Definitions

Ordinals are von Neumann ordinals, so an ordinal is the set of the ordinals below it and its order is membership. For a set $X$ of ordinals, $[X]^2$ is the set of two-element subsets of $X$. A coloring of $[X]^2$ with $n$ colors is a function $c:[X]^2\to\{0,\ldots,n-1\}$. A subset $H\subseteq X$ is homogeneous in color $i$ under $c$ when $c(p)=i$ for every $p\in[H]^2$. A set of ordinals carries the order inherited from the ordinals, and its order type $\operatorname{otp}(Y)$ is the unique ordinal order-isomorphic to it. A set of order type $3$ is a three-element set; when it is homogeneous in color $i$ it is a triangle of color $i$.

For an ordinal $\alpha$, ordinals $\beta_0,\ldots,\beta_{n-1}$ and an integer $n\ge1$, the relation

$$\alpha\to(\beta_0,\ldots,\beta_{n-1})^2_n$$

means: for every coloring $c:[\alpha]^2\to\{0,\ldots,n-1\}$ there are an index $i<n$ and a set $X\subseteq\alpha$ with $\operatorname{otp}(X)=\beta_i$ that is homogeneous in color $i$ under $c$. The subscript counts the colors and is omitted when $n=2$. When $\beta_1=\cdots=\beta_{n-1}=3$ the relation is written $\alpha\to(\beta_0,3,\ldots,3)^2_n$ and has $n-1$ triangle targets. This is the source's convention (p. 1) and the catalog's.

Two facts about the relation are used below without further comment.

1. Transport. Let $Y$ be a set of ordinals with $\operatorname{otp}(Y)=\alpha$ and suppose $\alpha\to(\beta_0,\ldots,\beta_{n-1})^2_n$. Then every coloring $d$ of $[Y]^2$ with $n$ colors has an index $i<n$ and a set $H\subseteq Y$ with $\operatorname{otp}(H)=\beta_i$ homogeneous in color $i$ under $d$. Proof: let $\pi:\alpha\to Y$ be the order isomorphism and color $[\alpha]^2$ by $d'(\{\xi,\eta\})=d(\{\pi(\xi),\pi(\eta)\})$; the relation gives $i<n$ and $X\subseteq\alpha$ of order type $\beta_i$ with $d'$ constantly $i$ on $[X]^2$; put $H=\pi[X]$. Since $\pi$ preserves order, $\operatorname{otp}(H)=\operatorname{otp}(X)=\beta_i$, and every pair in $[H]^2$ is the image of a pair in $[X]^2$, so $d$ is constantly $i$ on $[H]^2$.
2. Relabeling. The relation is unchanged by a bijective renaming of the colors that carries the list of targets along with it.

Statement

Let $\alpha$ be an ordinal with $\alpha\to(\alpha,3)^2$. Then for every finite $k\ge1$,

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

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

Proof

Induction on $k\ge1$. Write $P(k)$ for the displayed relation with $k$ triangle targets.

Base case. $P(1)$ is $\alpha\to(\alpha,3)^2_2$, which is the hypothesis.

Induction step. Assume $P(k)$ for some $k\ge1$. To prove $P(k+1)$, let $c:[\alpha]^2\to\{0,1,\ldots,k+1\}$ be any coloring with $k+2$ colors.

Merging two colors. Define $c':[\alpha]^2\to\{0,1,\ldots,k\}$ by

$$c'(p)= \begin{cases} 0 & \text{if } c(p)\in\{0,1\},\\ c(p)-1 & \text{if } c(p)\in\{2,\ldots,k+1\}. \end{cases}$$

So $c'$ merges the colors $0$ and $1$ of $c$ into the color $0$ and renames each color $j\in\{2,\ldots,k+1\}$ as $j-1\in\{1,\ldots,k\}$. The source writes the merged color $0'$ and keeps the names $2,\ldots,k+1$; the renaming here is the relabeling of fact 2 and changes nothing. The coloring $c'$ uses $k+1$ colors, so $P(k)$ applies to it, and one of the following two alternatives holds.

Alternative 1: a triangle under $c'$. There are $i\in\{1,\ldots,k\}$ and a three-element set $T\subseteq\alpha$ with $c'(p)=i$ for every $p\in[T]^2$. Since $i\ge1$, the definition of $c'$ forces $c(p)=i+1\in\{2,\ldots,k+1\}$ for every $p\in[T]^2$. Hence $T$ is a triangle of color $i+1$ under $c$, and $i+1\in\{1,\ldots,k+1\}$ is one of the triangle colors of $P(k+1)$.

Alternative 2: a large set under $c'$. There is $Y\subseteq\alpha$ with $\operatorname{otp}(Y)=\alpha$ and $c'(p)=0$ for every $p\in[Y]^2$. By the definition of $c'$, $c(p)\in\{0,1\}$ for every $p\in[Y]^2$, so $d=c\upharpoonright[Y]^2$ is a coloring of $[Y]^2$ with two colors. Since $\operatorname{otp}(Y)=\alpha$ and $\alpha\to(\alpha,3)^2$, fact 1 gives either a set $Z\subseteq Y$ with $\operatorname{otp}(Z)=\alpha$ and $d$ constantly $0$ on $[Z]^2$, or a three-element set $T\subseteq Y$ with $d$ constantly $1$ on $[T]^2$. As $d$ agrees with $c$ on $[Y]^2$, in the first case $Z$ is a subset of $\alpha$ of order type $\alpha$ homogeneous in color $0$ under $c$, and in the second case $T$ is a triangle of color $1$ under $c$, with $1\in\{1,\ldots,k+1\}$.

In every case $c$ has a subset of order type $\alpha$ homogeneous in color $0$ or a triangle of some color in $\{1,\ldots,k+1\}$. The coloring $c$ was arbitrary, so $P(k+1)$ holds. This completes the induction and proves the lemma.

Checks and scope

• Where the hypothesis is used. The relation $\alpha\to(\alpha,3)^2$ is used once in each induction step, in alternative 2, and only on a set of order type exactly $\alpha$. This is why the large target must equal the ambient ordinal: from a relation $\alpha\to(\beta,3)^2$ with $\beta<\alpha$ the induction hypothesis would return a set $Y$ of order type $\beta$, and the hypothesis would not apply to $Y$.
• Color counts. $c'$ has $k+1$ colors, and $P(k)$ speaks about colorings with $k+1$ colors; the triangle produced in alternative 1 has a color in $\{2,\ldots,k+1\}$ under $c$, the one in alternative 2 has color $1$, and together these are exactly the $k+1$ triangle colors of $P(k+1)$.
• Coverage. Every deduction of the source's proof (pp. 2--3) is written above. The reconstruction adds the transport argument of fact 1 and the explicit renaming of the colors, both of which the source leaves implicit; nothing is omitted.
• What the number $3$ contributes. Nothing beyond being a fixed target: the induction never uses that a triangle has three points. The source does not state this; Remark 3.2 (p. 4) only restates the lemma as the stability of $\alpha\to(\alpha,3)^2$ under adding finitely many colors with target $3$.

Depends on. Nothing beyond the hypothesis $\alpha\to(\alpha,3)^2$.

Consumed by. Theorem 3.1, with $\alpha=\omega_1\omega$.