Ordinal Ramsey: \(\omega_{3} → (\omega_{2} + 2)^{3}_{\omega}\)

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 \(\omega_3 \rightarrow (\omega_2+2)_{\omega}^3\)?

Baumgartner, Hajnal and Todorcevic [2] showed that GCH implies \( \omega_3 \rightarrow (\omega_2+\chi)_{k}^3\) for \( \chi < \omega_1\) and \( k < \omega\).

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 J. E. Baumgartner, A. Hajnal and S. Todorcevic, Extension of the Erdős-Rado theorems, Finite and infinite combinatorics in sets and logic (Banff, AB, 1991), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 411, 1-17, Kluwer Acad. Publ., Dordrecht, 1993.