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Combinatorica

, Volume 4, Issue 1, pp 71–78 | Cite as

Sparse ramsey graphs

  • Jaroslav Nešetřil
  • Vojtěch Rödl
Article

Abstract

IfH is a Ramsey graph for a graphG thenH is rich in copies of the graphG. Here we prove theorems in the opposite direction. We find examples ofH such that copies ofG do not form short cycles inH. This provides a strenghtening also, of the following well-known result of Erdős: there exist graphs with high chromatic number and no short cycles. In particular, we solve a problem of J. Spencer.

AMS subject classification (1980)

05 C 55 05 C 38 05 C 65 

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References

  1. [1]
    P. Erdős andA. Hajnal, On chromatic number of graphs and set systems,Acta Math. Acad. Sci. Hung. 17 (1966), 61–99.CrossRefGoogle Scholar
  2. [2]
    J. Folkman, Graphs with monochromatic complete subgraphs in every edge coloring,SIAM J. Appl. Math. 18 (1970), 19–29.zbMATHCrossRefMathSciNetGoogle Scholar
  3. [3]
    L. Lovász, On chromatic number of finite set-systems,Acta Math. Acad. Sci. Hungar. 19 (1968), 59–67.zbMATHCrossRefMathSciNetGoogle Scholar
  4. [4]
    J. Nešetřil andV. Rödl, Ramsey property of graphs with forbidden complete subgraphs,J. Comb. Th. (B) 20 (1976), 243–249.CrossRefGoogle Scholar
  5. [5]
    J. Nešetřil andV. Rödl, A short proof of the existence of highly chromatic hypergraphs without short cycles,J. Comb. Th. (B) 27 (1979), 225–227.CrossRefGoogle Scholar
  6. [6]
    J. Nešetřil andV. Rödl, Simple proof of the existence of restricted Ramsey graphs by means of a partite construction,Combinatorica 1 (1982), 199–202.Google Scholar
  7. [7]
    J. Nešetřil andV. Rödl,to appear.Google Scholar

Copyright information

© Akadémiai Kiadó 1984

Authors and Affiliations

  • Jaroslav Nešetřil
    • 1
  • Vojtěch Rödl
    • 2
  1. 1.Department of CyberneticsCharles UniversityPraha 1Czechoslovakia
  2. 2.FJFI ČVUT Department of MathematicsPraha 1Czechoslovakia

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