A GCH example of an ordinal graph with no infinite path
Author:
Jean A. Larson
Journal:
Trans. Amer. Math. Soc. 303 (1987), 383393
MSC:
Primary 03E50; Secondary 03E05, 04A20, 05C99
MathSciNet review:
896028
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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 satisfy a positive partition relation that can be expressed in graph theoretic terms. In symbols one writes to mean that every graph on an ordinal either has a subset order isomorphic to in which no two points are joined by an edge or has an infinite path. This positive result generalizes to ordinals of cardinality for a natural number. However, the argument, based on a set mapping theorem, works only on the initial segment of the limit ordinals of cardinality 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 , where is a natural number at least two.
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J. Larson, Martin's Axiom and ordinal graphs: large independent sets or infinite paths.
 [1]
 J. Baumgartner and J. Larson, A diamond example of an ordinal graph with no infinite paths.
 [2]
 T. Carlson, The pinup conjecture, Axiomatic Set Theory, Contemp. Math., vol. 31, Amer. Math. Soc., Providence, R.I., 1984, pp. 4162. 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, NorthHolland, Amsterdam, 1969, pp. 327363. MR 0299537 (45:8585)
 [4]
 J. Gregory, Higher Souslin trees and the Generalized Continuum Hypothesis, J. Symbolic Logic 41 (1976), 663671. 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.
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947198708960286
PII:
S 00029947(1987)08960286
Article copyright:
© Copyright 1987
American Mathematical Society
