Ordinal Ramsey: \(\omega_{1}^{2} → (\omega_{1}^{2}, 3)^{2}\)
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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
\( 1 \leq r,k < \omega.\)
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.