Ordinal Ramsey: If \(\alpha → (\alpha, 3)^{2},\) then \(\alpha → (\alpha, 4)^{2}\)
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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
\( 1 \leq r,k < \omega.\)
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For the case of \( n=4\), Darby and Larson (unpublished) proved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
\( \omega^{\omega^{2}}
\rightarrow ( \omega^{\omega^{2}},3)^2\).