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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: 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
##### Keywords:
sigma trees; arrow partition relations
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