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A GCH example of an ordinal graph with no infinite path. (English) Zbl 0638.05002

Let \(\alpha\to (\alpha\), infinite path) denote the fact 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. P. Erdős, A. Hajnal and E. C. Milner [Combinat. Theory Appl. Colloquia Math. Soc. János Bolyai 4, 327-363 (1969; Zbl 0215.329)] have proved that limit ordinals \(\alpha <\omega_ 1^{\omega +2}\) satisfy this partition relation \(\alpha\to (\alpha\), infinite path). In the paper under review, the author proves the following theorem: Assume the generalized continuum hypothesis. For every positive integer \(n\geq 2\), there is a cofinal set of ordinals \(\alpha <\omega_ n\) so that \(\alpha \nrightarrow (\alpha\), infinite path).
Reviewer: E.Fuchs

MSC:

05A05 Permutations, words, matrices
03E50 Continuum hypothesis and Martin’s axiom
03E05 Other combinatorial set theory
05A17 Combinatorial aspects of partitions of integers
05C38 Paths and cycles

Citations:

Zbl 0215.329
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References:

[1] J. Baumgartner and J. Larson, A diamond example of an ordinal graph with no infinite paths. · Zbl 0703.03028
[2] Tim Carlson, The pin-up conjecture, Axiomatic set theory (Boulder, Colo., 1983) Contemp. Math., vol. 31, Amer. Math. Soc., Providence, RI, 1984, pp. 41 – 62. · Zbl 0557.03029 · doi:10.1090/conm/031/763892
[3] P. Erdős, A. Hajnal, and E. C. Milner, Set mappings and polarized partition relations, Combinatorial theory and its applications, I (Proc. Colloq., Balatonfüred, 1969) North-Holland, Amsterdam, 1970, pp. 327 – 363. · Zbl 0215.32903
[4] John Gregory, Higher Souslin trees and the generalized continuum hypothesis, J. Symbolic Logic 41 (1976), no. 3, 663 – 671. · Zbl 0347.02044 · doi:10.2307/2272043
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[6] J. Larson, Martin’s Axiom and ordinal graphs: large independent sets or infinite paths. · Zbl 0703.03029
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