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For any set $X$ and cardinal $\mu \neq \emptyset$, we denote by $[X]^\mu$ the collection of subsets of cardinality $\mu$. If $\kappa, \mu \neq \emptyset$ are cardinals and $f: [X]^\mu\to \kappa$ is a map, we say that $H\subseteq X$ is homogeneous with respect to $f$ if the restriction $f|_{H^\mu}: H^\mu \to \kappa$ is constant.

For cardinals $\lambda, \mu, \kappa\neq \emptyset$ and any set $X\neq \emptyset$ we write $$X \to (\lambda)^\mu_\kappa$$ if for every map $f: [X]^\mu\to\kappa$ there is $H\subseteq X$ such that $H$ is homogeneous with respect to $f$ and $|H|=\lambda$.

With the help of the Axiom of Choice ${\sf (AC)}$ one can prove that $X \not\to (\omega)^\omega_2$ for every infinite $X$ (see Theorem 7, p. 5 of this recommended introduction to infinite combinatorics, thank you to Burak for writing it!).

Question. Does the statement "$X \not\to (\omega)^\omega_2$ for every infinite set $X$" imply ${\sf (AC)}$?

For any set $X$ and cardinal $\mu \neq \emptyset$, we denote by $[X]^\mu$ the collection of subsets of cardinality $\mu$. If $\kappa, \mu \neq \emptyset$ are cardinals and $f: [X]^\mu\to \kappa$ is a map, we say that $H\subseteq X$ is homogeneous with respect to $f$ if the restriction $f|_{[H]^\mu}: [H]^\mu \to \kappa$ is constant.

For cardinals $\lambda, \mu, \kappa\neq \emptyset$ and any set $X\neq \emptyset$ we write $$X \to (\lambda)^\mu_\kappa$$ if for every map $f: [X]^\mu\to\kappa$ there is $H\subseteq X$ such that $H$ is homogeneous with respect to $f$ and $|H|=\lambda$.

With the help of the Axiom of Choice ${\sf (AC)}$ one can prove that $X \not\to (\omega)^\omega_2$ for every infinite $X$ (see Theorem 7, p. 5 of this recommended introduction to infinite combinatorics, thank you to Burak for writing it!).

Question. Does the statement "$X \not\to (\omega)^\omega_2$ for every infinite set $X$" imply ${\sf (AC)}$?

For any set $X$ and cardinal $\mu \neq \emptyset$, we denote by $[X]^\mu$ the collection of subsets of cardinality $\mu$. If $\kappa, \mu \neq \emptyset$ are cardinals and $f: [X]^\mu\to \kappa$ is a map, we say that $H\subseteq X$ is homogeneous with respect to $f$ if the restriction $f|_{H^\mu}: H^\mu \to \kappa$ is constant.

For cardinals $\lambda, \mu, \kappa\neq \emptyset$ and any set $X\neq \emptyset$ we write $$X \to (\lambda)^\mu_\kappa$$ if for every map $f: [X]^\mu\to\kappa$ there is $H\subseteq X$ such that $H$ is homogeneous with respect to $f$ and $|H|=\lambda$.

With the help of the Axiom of Choice ${\sf (AC)}$ one can prove that $X \not\to (\omega)^\omega_2$ for every infinite $X$ (see Theorem 7, p. 5 of this introduction to infinite combinatorics).

Question. Does the statement "$X \not\to (\omega)^\omega_2$ for every infinite set $X$" imply ${\sf (AC)}$?

For any set $X$ and cardinal $\mu \neq \emptyset$, we denote by $[X]^\mu$ the collection of subsets of cardinality $\mu$. If $\kappa, \mu \neq \emptyset$ are cardinals and $f: [X]^\mu\to \kappa$ is a map, we say that $H\subseteq X$ is homogeneous with respect to $f$ if the restriction $f|_{H^\mu}: H^\mu \to \kappa$ is constant.

For cardinals $\lambda, \mu, \kappa\neq \emptyset$ and any set $X\neq \emptyset$ we write $$X \to (\lambda)^\mu_\kappa$$ if for every map $f: [X]^\mu\to\kappa$ there is $H\subseteq X$ such that $H$ is homogeneous with respect to $f$ and $|H|=\lambda$.

With the help of the Axiom of Choice ${\sf (AC)}$ one can prove that $X \not\to (\omega)^\omega_2$ for every infinite $X$ (see Theorem 7, p. 5 of this recommended introduction to infinite combinatorics, thank you to Burak for writing it!).

Question. Does the statement "$X \not\to (\omega)^\omega_2$ for every infinite set $X$" imply ${\sf (AC)}$?

For any set $X$ and cardinal $\mu \neq \emptyset$, we denote by $[X]^\mu$ the collection of subsets of cardinality $\mu$. If $\kappa, \mu \neq \emptyset$ are cardinals and $f: [X]^\mu\to \kappa$ is a map, we say that $H\subseteq X$ is homogeneous with respect to $f$ if the restriction $f|_{H^\mu}: H^\mu \to \kappa$ is constant.

For cardinals $\lambda, \mu, \kappa\neq \emptyset$ and any set $X\neq \emptyset$ we write $$X \to (\lambda)^\mu_\kappa$$ if for every map $f: [X]^\mu\to\kappa$ there is $H\subseteq X$ such that $H$ is homogeneous with respect to $f$ and $|H|=\lambda$.

With the help of the Axiom of Choice ${\sf (AC)}$ one can prove that $X \not\to (\omega)^\omega_2$ for every infinite $X$ (see Theorem 7, p. 5 of this introduction to infinite combinatorics.)

Question. Does the statement that $X \not\to (\omega)^\omega_2$ for every infinite $X$ imply ${\sf (AC)}$?

For any set $X$ and cardinal $\mu \neq \emptyset$, we denote by $[X]^\mu$ the collection of subsets of cardinality $\mu$. If $\kappa, \mu \neq \emptyset$ are cardinals and $f: [X]^\mu\to \kappa$ is a map, we say that $H\subseteq X$ is homogeneous with respect to $f$ if the restriction $f|_{H^\mu}: H^\mu \to \kappa$ is constant.

For cardinals $\lambda, \mu, \kappa\neq \emptyset$ and any set $X\neq \emptyset$ we write $$X \to (\lambda)^\mu_\kappa$$ if for every map $f: [X]^\mu\to\kappa$ there is $H\subseteq X$ such that $H$ is homogeneous with respect to $f$ and $|H|=\lambda$.

With the help of the Axiom of Choice ${\sf (AC)}$ one can prove that $X \not\to (\omega)^\omega_2$ for every infinite $X$ (see Theorem 7, p. 5 of this introduction to infinite combinatorics).

Question. Does the statement "$X \not\to (\omega)^\omega_2$ for every infinite set $X$" imply ${\sf (AC)}$?