Ordinal Ramsey: \(\omega_{1} → (\alpha, 4)^{3}\) for \(\alpha < \omega_{1}\)

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 \( 1 \leq r,k < \omega.\)

Problem [1]

Is it true that \( \omega_1 \rightarrow (\alpha,4)^3\) for \(\alpha < \omega_1\)?

Milner and Prikry [2] gave an affirmative answer for \( \alpha \leq \omega^2 + 1\).

Bibliography
1 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.

2 E. C. Milner and K. Prikry, A partition relation for triples using a model of Todor\({\v{c\/}}\kern.05em\)evi\({\'{c\/}}\), Directions in Infinite Graph Theory and Combinatorics, Proceedings International Conference (Cambridge, 1989), Discrete Math. 95 (1991), 183-191.