Ordinal Ramsey: \(\omega_{1}^{2} → (\omega_{1}^{2}, 3)^{2}\)

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 [2]

Is it true that \({\omega_1}^2 \rightarrow ({\omega_1}^2,3)^2 \)?

A. Hajnal [5] proved \( {\omega_1}^2\not\rightarrow ({\omega_1}^2,3)^2\) under CH.

Erdös and Hajnal [3] ask if \( \hbox{MA}_{\aleph_1}\) + \( 2^{\aleph_0}=\aleph_2\) implies \( {\omega_1}^2 \rightarrow ({\omega_1}^2,3)^2 \)?

Erdös, Hajnal and Larson [4] asked for the cardinals \( \lambda\) that \( \lambda^2\rightarrow (\lambda^2,3)^2\) holds. Hajnal [5] showed the relation failed at successors of regular cardinals under GCH. Baumgartner [1] showed that the relation failed at successors of singular cardinals under GCH.


Bibliography
1 J. E. Baumgartner, Partition relations for uncountable ordinals, Israel J. Math. 21 (1975), 296-307.

2 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.

3 P. Erdős and A. Hajnal, Unsolved and solved problems in set theory, Proceedings of the Tarski Symposium (Proc. Sympos. Pure Math., Vol. XXV, Univ. California, Berkeley, Calif., 1971), pp. 269-287, Amer. Math. Soc., Providence, R. I., 1974.

4 P. Erdős, A. Hajnal and J. Larson, Ordinal partition behavior of finite powers of cardinals, Finite and infinite combinatorics in sets and logic (Banff, AB, 1991), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 411, 97-115, Kluwer Acad. Publ., Dordrecht, 1993.

5 A. Hajnal, A negative partition relation, Proc. Nat. Acad. Sci. USA 68 (1971),142-144.