Ordinal Ramsey: For which $\alpha$ does $\omega^{\alpha} → (\omega^{\alpha}, 3)^{2}$?

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Here we use the following arrow notation, first introduced by Rado:

$\displaystyle \kappa \rightarrow ( \lambda_{\nu})_{\gamma}^r $

which means that for any function $ f: [\kappa]^r \rightarrow \gamma$ there are $ \nu < \gamma$ and $ H \subset \kappa$ such that $ H$ has order type $ \lambda_{\nu} $ and $ f(Y) = \nu$ for all $ Y \in [H]^r$ (where $ [H]^r$ denotes the set of $ r$-element subsets of $ H$). If $ \lambda_{\nu}=\lambda$ for all $ \nu < \gamma$, then we write $ \kappa\rightarrow (\lambda)_{\gamma}^r$. In this language, Ramsey's theorem can be written as

$\displaystyle \omega \rightarrow (\omega)_k^r $

for $ 1 \leq r,k < \omega.$

A conjecture on ordinary partition relations for ordinals ($1000, proposed by Erdös and Hajnal [3])

Determine the $\alpha$'s for which $\omega^{\alpha} \rightarrow (\omega^{\alpha},3)^2$.

Galvin and Larson [4] showed that such $ \alpha$ must be of the form $ \omega^{\beta}$. Chang [1] proved $ \omega^{\omega} \rightarrow (\omega^{\omega},3)^2$. Milner [7] generalized the proof of Chang to show $ \omega^{\omega} \rightarrow (\omega^{\omega},n)^2$ for $ n < \omega$, and Larson [5] gave a simpler proof.

There have been many recent developments on ordinary partition relations for countable ordinals. Schipperus [8] proved that, if $ \beta$ is the sum of at most two indecomposables, then

\begin{equation} \omega^{\omega^{\beta}} \rightarrow ( \omega^{\omega^{\beta}},3)^2. \end{equation}

In the other direction, Schipperus [8] and Larson [6] showed that, if $ \beta$ is the sum of two indecomposables, then

\begin{equation} \omega^{\omega^{\beta}} \not \rightarrow ( \omega^{\omega^{\beta}},5)^2. \end{equation}

Darby [2] proved that, if $ \beta$ is the sum of three indecomposables, then

\begin{equation} \omega^{\omega^{\beta}} \not \rightarrow ( \omega^{\omega^{\beta}},4)^2. \end{equation}

Schipperus [8] also proved that, if $ \beta$ is the sum of four indecomposables, then

\begin{equation} \omega^{\omega^{\beta}} \not \rightarrow ( \omega^{\omega^{\beta}},3)^2. \end{equation}

Bibliography
1	C. C. Chang, A partition theorem for the complete graph on $ \omega^{\omega}$, J. Comb. Theory Ser. A 12 (1972), 396-452.
2	C. Darby, Negative partition relations for ordinals $ \omega^{\omega^{\alpha}}$, J. Comb. Theory Ser. B 76 (1999), 205-222.
3	P. Erdős and A. Hajnal, Unsolved problems in set theory, Axiomatic Set Theory (Proc. Sympos. Pure Math., Vol. XIII, Part I, Univ. California, Los Angeles, Calif,. 1967), 17-48, Amer. Math. Soc., Providence, R. I., 1971.
4	F. Galvin and J. Larson, Pinning countable ordinals, Fund. Math. 82 (1974/75), 357-361.
5	J. A. Larson, A short proof of a partition theorem for the ordinal $ \omega^{\omega}$, Ann. Math. Logic 6 (1973/74), 129-145.
6	J. A. Larson, An ordinal partition avoiding pentagrams, preprint.
7	E. C. Milner, Lecture notes on partition relations for ordinal numbers (1972), unpublished.
8	Rene Schipperus, Countable partition ordinals, Ph. D. thesis, Univ. of Calgary. Published version: Annals of Pure and Applied Logic 161 (2010), 1195-1215.

Ordinal Ramsey: For which \(\alpha\) does \(\omega^{\alpha} → (\omega^{\alpha}, 3)^{2}\)?

A conjecture on ordinary partition relations for ordinals ($1000, proposed by Erdös and Hajnal [3])