Skip to main content
SpringerLink
Log in

Some extremal problems for simple two-complexes

  • Conference paper
  • First Online:
Theory and Applications of Graphs

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 642))

  • 1513 Accesses

Abstract

A variety of extremal problems are considered for the members of two particular families of simple 2-complexes. We let Bk denote the complex consisting of k 2-cells all sharing one single edge in common and Wk the complex consisting of k 2-cells sharing exactly one vertex in common. The problems treated here include the following questions for the Bk and the corresponding questions for the Wk.

Suppose that for each 2-coloring of the 2-cells of a complex G, either one color contains a monochromatic copy of Bk or the other contains a monochromatic copy of Bℓ. How many 2-cells must G have? How many vertices? How many vertices and how many 2-cells must G have if one of Bk and Bℓ must be "faithfully imbedded" in one of the colors? How do these numbers compare with the "generalized Ramsey numbers" for the pairs (Bk, Bℓ)? What is the smallest integer f(n, k) such that each 2-complex on n vertices having f(n, k) 2-cells contains a copy of Bk?

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 44.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 59.00
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. W. Brown, P. Erdős, and V. T. Sos, On the existence of triangulated spheres in 3-graphs and related problems. Period, Math. Hungar. 3(1973) 221–228.

    Article  MathSciNet  MATH  Google Scholar 

  2. J. Doyen, Constructions of disjoint Steiner triple systems. Proc. Amer. Math. Soc. 32(1972) 409–416.

    Article  MathSciNet  MATH  Google Scholar 

  3. R. A. Duke and F. Harary, Generalized Ramsey Theory VI: Ramsey numbers for small plexes. J. Austral. Math. Soc., to appear.

    Google Scholar 

  4. R. A. Duke, Ramsey numbers of families of 2-complexes. Proc. of the Sixth Southeastern Conference on Combinatories, Graph Theory, and Computing. Congressus Numerantium No. XIV, Utilitas Math., Winnipeg, Man. (1975) 265–277.

    Google Scholar 

  5. P. Erdős, Extremal problems on graphs and hypergraphs. Hypergraph Seminar. Springer-Verlag, New York (1974) 75–84.

    Chapter  Google Scholar 

  6. P. Erdős, Problems and results on finite and infinite graphs, to appear.

    Google Scholar 

  7. P. Erdős, private communication.

    Google Scholar 

  8. P. Erdős, R. J. Faudree, R. H. Schelp and C. C. Rousseau, The size Ramsey number, to appear.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Georgia Institute of Technology, Atlanta, GA, 30332, USA

    Richard A. Duke

Authors
  1. Richard A. Duke
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Department of Mathematics, Western Michigan University, Kalamazoo, MI, 49008, USA

    Yousef Alavi  & Don R. Lick  & 

Rights and permissions

Reprints and permissions

Copyright information

© 1978 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Duke, R.A. (1978). Some extremal problems for simple two-complexes. In: Alavi, Y., Lick, D.R. (eds) Theory and Applications of Graphs. Lecture Notes in Mathematics, vol 642. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0070372

Download citation

  • DOI: https://doi.org/10.1007/BFb0070372

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-08666-6

  • Online ISBN: 978-3-540-35912-8

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation