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Graphs and Orders in Ramsey Theory and in Dimension Theory

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Book cover Graphs and Order

Part of the book series: NATO ASI Series ((ASIC,volume 147))

Abstract

The purpose of this paper is to present a concise and relatively self contained treatment of recent results linking partially ordered sets with topics more traditionally associated with graph theory and combinatorics: Ramsey theory and chromatic graph theory. In particular, we will present the major theorems of Nesetril and Rödl ([31], [33], and [34]) concerning Ramsey theory for partially ordered sets in a new setting which will allow nonspecialists to appreciate the power and beauty of these results. Other Ramsey theoretic results for partially ordered sets will be discussed briefly and some directions for future research will be indicated.

We will also present a concise treatment of the constructions of Ross and Trotter ([53], [54]) for irreducible partially ordered sets utilizing familiar concepts from chromatic graph (and hypergraph) theory. When combined with the complexity theorems of Yannakakis [57], these constructions show that partially ordered sets can simultaneously exhibit both mathematical elegance and awkward pathology.

1Research supported in part by NSF Grant ISP-80110451

2Research supported in part by NSF Grants MCS-8202172 and DMS-8401281

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Author information

Authors and Affiliations

  1. Department of Mathematics and Statistics, University of South Carolina, Columbia, S.C., 29208, USA

    M. Paoli, W. T. Trotter Jr & J. W. Walker

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  1. M. Paoli
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  2. W. T. Trotter Jr
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  3. J. W. Walker
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Editors and Affiliations

  1. Department of Mathematics and Statistics, The University of Calgary, Calgary, Alberta, Canada

    Ivan Rival

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© 1985 D. Reidel Publishing Company

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Paoli, M., Trotter, W.T., Walker, J.W. (1985). Graphs and Orders in Ramsey Theory and in Dimension Theory. In: Rival, I. (eds) Graphs and Order. NATO ASI Series, vol 147. Springer, Dordrecht. https://doi.org/10.1007/978-94-009-5315-4_9

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  • DOI: https://doi.org/10.1007/978-94-009-5315-4_9

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-94-010-8848-0

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