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](https://users.metu.edu.tr/burakk/lecturenotes/village2017lecturenotes.pdf), thank you to Burak for writing it!).


**Question.** Does the statement "$X \not\to (\omega)^\omega\_2$ for every infinite set $X$" imply ${\sf (AC)}$?



Does "$X \not\to (\omega)^\omega_2$ for every infinite $X$" imply ${\sf AC}$?
The answer is no, the statement that for every set $X$ we have $$X\not\to(\omega)^\omega\_2$$ does not imply the axiom of choice.


This was shown by Kleinberg and Seiferas in 1973, see



> 
> [MR0340025](https://mathscinet.ams.org/mathscinet-getitem?mr=340025) (49 #4782)
> Kleinberg, E. M.; Seiferas, J. I.
> Infinite exponent partition relations and well-ordered choice.
> J. Symbolic Logic 38 (1973), 299–308.
> <https://doi.org/10.2307/2272066>
> 
> 
> 


For $\kappa$ a (well-ordered) infinite cardinal, $\kappa$-well-ordered choice, $\mathsf{AC}\_\kappa$, is the statement that every $\kappa$-sequence of nonempty sets admits a choice function.


The axiom of well-ordered choice $\mathsf{WOC}$ is the statement that $\mathsf{AC}\_\kappa$ holds for all infinite well-ordered $\kappa$.


This statement is strictly weaker than the axiom of choice: it does not imply that $\mathbb R$ is well-orderable, and even if we add this assumption, the result is still weaker than choice. See for instance theorem 5.1 in



> 
> [MR1351415](https://mathscinet.ams.org/mathscinet-getitem?mr=1351415) (96h:03087)
> Higasikawa, Masasi
> Partition principles and infinite sums of cardinal numbers.
> Notre Dame J. Formal Logic 36 (1995), no. 3, 425–434. <https://doi.org/10.1305/ndjfl/1040149358>
> 
> 
> 


However, as shown in the paper by Kleinberg and Seiferas, $\mathsf{WOC}$ plus the existence of a well-ordering of $[\omega]^\omega$ rules out all infinite exponent partition relations. It is still open (as far as I know) whether $\mathsf{WOC}$ suffices for this result. What Kleinberg and Seiferas show is that, under $\mathsf{WOC}$, either all infinite exponent partition relations fail, or else $\omega\to(\omega)^\omega\_2$. (And the latter fails if $[\omega]^\omega$ is well-orderable.)


