# zbMATH — the first resource for mathematics

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
##### Keywords:
sigma trees; arrow partition relations
Full Text:
##### References:
 [1] Baumgartner, J. &Hajnal, A., A proof (involving Martin’s axiom) of a partition relation.Fund. Math., 73 (1973), 193–203. · Zbl 0257.02054 [2] Baumgartner, J., Malitz, J. &Reinhardt, W., Embedding trees in the rationals.Proc. Nat. Acad. Sci. U.S.A., 67 (1970), 1748–1753. · Zbl 0209.01601 [3] Erdös, P. &Hajnal, A., Unsolved problems in set theory.Amer. Math. Soc. Proc. Symp. Pure Math., 13, Part 1 (1971), 17–48. · Zbl 0228.04001 [4] –, Unsolved and solved problems in set theory.Amer. Math. Soc. Proc. Symp. Pure Math., 25 (1974), 267–287. [5] Erdös, P., Hajnal, A., Máté, A. & Rado, R.,Combinatorial set theory: Partition relations for cardinals. North-Holland, 1984. · Zbl 0573.03019 [6] Erdös, P., Hajnal, A. &Rado, R., Partition relations for cardinal numbers.Acta Math. Acad. Sci. Hungar., 16 (1965), 93–196. · Zbl 0158.26603 [7] Erdös, P. &Rado, R., Combinatorial theorems on classifications of subsets of a given set.Proc. London Math. Soc., 2 (1952), 417–439. · Zbl 0048.28203 [8] –, A partition calculus in set theory.Bull. Amer. Math. Soc., 62 (1956), 427–489. · Zbl 0071.05105 [9] Galvin, F., Partition theorems for the real line.Notices Amer. Math. Soc., 15 (1968), 660; Erratum 16 (1969), 1095. [10] Galvin, F., On a partition theorem of Baumgartner and Hajnal.Coll. Math. Soc., János Bolyai, 10. Infinite and finite sets. Keszthely (Hungary), 1973, North-Holland (1975), 711–729. [11] Hajnal, A., Some results and problems on set theory.Acta Math. Acad. Sci. Hungary., 11 (1960), 277–298. · Zbl 0106.00901 [12] Hartogs, F., Über das Problem der Wohlordnung.Math. Ann., 76 (1915), 438–443. · JFM 45.0125.01 [13] Kurepa, D., Ensembles ordonnés et ramifiés.Publ. Math. Univ. Belgrade, 4 (1935), 1–138. · JFM 61.0980.01 [14] –, Ensembles ordonnés et leurs sous-ensembles bien ordonnés.C. R. Acad. Sci. Paris Ser. A, 242 (1956), 2202–2203. · Zbl 0071.05004 [15] –, Monotone mappings between some kinds of ordered sets.Glasnik Mat.-Fiz. Astr., 19 (1964), 175–186. · Zbl 0134.25602 [16] Laver, R., Partition relations for uncountable cardinals $$\leqslant 2^{\aleph _0 }$$ .Coll. Math. Soc. János Bolyai, 10. Infinite and finite sets. Keszthely (Hungary), 1973, North-Holland (1975), 1029–1042. [17] Martin, D. A. &Solovay, R. M., Internal Cohen extensions.Ann. Math. Logic., 2 (1970), 143–178. · Zbl 0222.02075 [18] Máté, A., A partition relation for Souslin trees,Trans. Amer. Math. Soc., 261 (1981), 143–149. · Zbl 0497.03038 [19] Nash-Williams, C. St. J. A., On well-quasi-ordering transfinite sequences.Proc. Cambridge Phil. Soc., 61 (1965), 33–39. · Zbl 0129.00602 [20] Neumer, J., Verallgemeinerung eines Satzes von Alexandroff und Urysohn.Math. Z., 54 (1951), 254–261. · Zbl 0042.28103 [21] Prikry, K., On a set-theoretic partition problem.Duke Math. J., 39 (1972), 77–83. · Zbl 0239.04004 [22] Shelah, S., Notes on combinatorial set theory.Israel J. Math., 14 (1973), 262–277. · Zbl 0269.04004 [23] Tarski, A., On well-ordered subsets of any set.Fund. Math., 32 (1939), 176–183. · Zbl 0021.11003 [24] Todorcevic, S., Stationary sets, trees and continuums.Publ. Inst. Math. Beograd, 27 (41) (1981), 249–262. · Zbl 0519.06002 [25] –, Real functions on the family of all well-ordered subsets of a partially ordered set.J. Symbolic Logic, 48 (1983), 91–96. · Zbl 0508.06002 [26] –, A partition relation for partially ordered sets.Abstracts Amer. Math. Soc., 2 (1981), A-362. [27] –, Forcing positive partition relations.Trans. Amer. Math. Soc., 280 (1983), 703–720. · Zbl 0532.03023
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.