Larson, Jean A. A GCH example of an ordinal graph with no infinite path. (English) Zbl 0638.05002 Trans. Am. Math. Soc. 303, 383-393 (1987). 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 Cited in 1 ReviewCited in 1 Document 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 Keywords:ordinal graph; infinite path; generalized continuum hypothesis Citations:Zbl 0215.329 PDFBibTeX XMLCite \textit{J. A. Larson}, Trans. Am. Math. Soc. 303, 383--393 (1987; Zbl 0638.05002) Full Text: DOI 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 [5] Thomas Jech, Set theory, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1978. Pure and Applied Mathematics. · Zbl 0419.03028 [6] J. Larson, Martin’s Axiom and ordinal graphs: large independent sets or infinite paths. · Zbl 0703.03029 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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.