Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS

Mobile Device Pairing
Green Open Access
Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)


A GCH example of an ordinal graph with no infinite path

Author: Jean A. Larson
Journal: Trans. Amer. Math. Soc. 303 (1987), 383-393
MSC: Primary 03E50; Secondary 03E05, 04A20, 05C99
MathSciNet review: 896028
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: It is hard to find nontrivial positive partition relations which hold for many ordinals in ordinary set theory, or even ordinary set theory with the additional assumption of the Generalized Continuum Hypothesis. Erdös, Hajnal and Milner have proved that limit ordinals $ \alpha < \omega _1^{\omega + 2}$ satisfy a positive partition relation that can be expressed in graph theoretic terms. In symbols one writes $ \alpha \to {(\alpha ,\,\operatorname{infinite} \operatorname{path} )^2}$ to mean that every graph on an ordinal $ \alpha $ either has a subset order isomorphic to $ \alpha $ in which no two points are joined by an edge or has an infinite path. This positive result generalizes to ordinals of cardinality $ {\aleph _m}$ for $ m$ a natural number. However, the argument, based on a set mapping theorem, works only on the initial segment of the limit ordinals of cardinality $ {\aleph _m}$ for which the set mapping theorem is true. In this paper, the Generalized Continuum Hypothesis is used to construct counterexamples for a cofinal set of ordinals of cardinality $ {\aleph _m}$, where $ m$ is a natural number at least two.

References [Enhancements On Off] (What's this?)

  • [1] J. Baumgartner and J. Larson, A diamond example of an ordinal graph with no infinite paths.
  • [2] T. Carlson, The pin-up conjecture, Axiomatic Set Theory, Contemp. Math., vol. 31, Amer. Math. Soc., Providence, R.I., 1984, pp. 41-62. MR 763892 (86b:03059)
  • [3] P. Erdös, A. Hajnal and E. C. Milner, Set mappings and polarized partition relations, Combinatorial Theory and its Applications, Balatonfüred, Colloq. Math. Soc. János Bolyai, vol. 4, North-Holland, Amsterdam, 1969, pp. 327-363. MR 0299537 (45:8585)
  • [4] J. Gregory, Higher Souslin trees and the Generalized Continuum Hypothesis, J. Symbolic Logic 41 (1976), 663-671. MR 0485361 (58:5208)
  • [5] T. Jech, Set theory, Academic Press, New York, 1978. MR 506523 (80a:03062)
  • [6] J. Larson, Martin's Axiom and ordinal graphs: large independent sets or infinite paths.

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 03E50, 03E05, 04A20, 05C99

Retrieve articles in all journals with MSC: 03E50, 03E05, 04A20, 05C99

Additional Information

PII: S 0002-9947(1987)0896028-6
Article copyright: © Copyright 1987 American Mathematical Society