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Partition relations for partially ordered sets. (English) Zbl 0603.03013
The main aim of the author is to study arrow partition relations concerning partially ordered sets P such that (1) \(P\to (\kappa)^ 1_{\kappa}\) for some infinite cardinal \(\kappa\). For \(\kappa =\omega\), simple examples of such P’s are \(\omega_ 1\) and R. If P is a tree T, then \(\neg (1)\) is equivalent to the statement that T is union of \(\leq \kappa\) antichains because any partition F of T yields a refined partition of T: the system of all rows of all members of F.
Theorem 1 (conjectured by F. Galvin): (1) implies \(P\to (\alpha)^ 2_{\kappa}\) for all \(\alpha <\omega_ 1\) and \(\kappa <\omega\). (Theorem 1 strengthens several results obtained earlier by P. Erdős and R. Rado, A. Hajnal, F. Galvin, K. Prikry, J. Baumgartner; e.g., Galvin’s result is obtained from Theorem 1 writing \((\eta)\) instead of \((\omega)\).)
Theorem 2: Let \(\kappa\) be a regular cardinal and \(\lambda\) an infinite cardinal such that \(\lambda^{{\underset \smallsmile \lambda}}<\kappa\); if \(P\to (\kappa)^ 1_{2^{{\underset \smallsmile \kappa}}}\), then \(P\to (\kappa +\xi)^ 2_ 2\) for all \(\xi <\lambda.\)
Theorem 4[3]: Let \(\lambda \geq \aleph_ 0\), \(\theta\geq 2\) and (1) for \(\kappa:=\theta^{{\smallsmile\to \lambda}}\); then \(P\to (\lambda +1)_{\theta}^{<\omega}\) \([P\to (\alpha,(cf \lambda +1)_{\gamma})^ 2\) for \(\alpha<\kappa^+\) and \(\gamma <cf \lambda].\)
Theorem 9: The following are equivalent:
\(\neg (1).\)
There is a strictly increasing mapping of the tree \(\sigma P\) into \((P(\kappa),\subset).\)
The tree \(\sigma'P\) is union of \(\leq \kappa\) antichains (\(\sigma P\) denotes the tree of all well-ordered bounded subsets of P, ordered by the relation ”is an initial segment of”, \(\sigma'T:=\{x|\) \(x\in \sigma T\), \(tp\;x-1<tp\;x\}).\)
A number of further theorems and corollaries are also proved. The proofs of the main theorems 1-4 are based on tree considerations connected with Theorem 9.
Reviewer: D.Kurepa

MSC:
03E05 Other combinatorial set theory
05C55 Generalized Ramsey theory
06A06 Partial orders, general
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