Ordinal Ramsey: If \(\alpha → (\alpha, 3)^{2},\) then \(\alpha → (\alpha, 4)^{2}\)

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

A problem on ordinary partition relations for ordinals (proposed by Erdös and Hajnal [1])

Is it true that if \(\alpha \rightarrow (\alpha,3)^2\), then \(\alpha \rightarrow (\alpha,4)^2\)?

The original problem proposed in [1] was ``Is it true that if \( \alpha \rightarrow (\alpha,3)^2\), then \( \alpha \rightarrow (\alpha,n)^2\)?". However, Schipperus' results [2] (see this problem for more) give a negative answer for the case of \( n \geq 5\).

For the case of \( n=4\), Darby and Larson (unpublished) proved

\(\displaystyle \omega^{\omega^{2}} \rightarrow ( \omega^{\omega^{2}},4)^2 \)

extending the previous work of Darby on \( \omega^{\omega^{2}} \rightarrow ( \omega^{\omega^{2}},3)^2\).


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 Rene Schipperus, Countable partition ordinals, Ph. D. thesis, Univ. of Calgary. Published version: Annals of Pure and Applied Logic 161 (2010), 1195-1215.