Ordinal Ramsey: \(\omega_{1} → (\alpha, 4)^{3}\) for \(\alpha < \omega_{1}\)
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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 [1]
Is it true that \( \omega_1 \rightarrow (\alpha,4)^3\) for \(\alpha < \omega_1\)?Milner and Prikry [2] gave an affirmative answer for \( \alpha \leq \omega^2 + 1\).
Bibliography | |
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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 |
E. C. Milner and K. Prikry,
A partition relation for triples using a model of Todor\({\v{c\/}}\kern.05em\)evi\({\'{c\/}}\),
Directions in Infinite Graph Theory and Combinatorics,
Proceedings International Conference (Cambridge, 1989),
Discrete Math. 95 (1991), 183-191.
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