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Graphs and Combinatorics

, Volume 2, Issue 1, pp 135–144 | Cite as

Large triangle-free subgraphs in graphs withoutK4

  • P. Frankl
  • V. Rödl
Article

Abstract

It is shown that for arbitrary positiveε there exists a graph withoutK4 and so that all its subgraphs containing more than 1/2 +ε portion of its edges contain a triangle (Theorem 2). This solves a problem of Erdös and Nešetřil. On the other hand it is proved that such graphs have necessarily low edge density (Theorem 4).

Theorem 3 which is needed for the proof of Theorem 2 is an analog of Goodman's theorem [8], it shows that random graphs behave in some respect as sparse complete graphs.

Theorem 5 shows the existence of a graph on less than 1012 vertices, withoutK4 and which is edge-Ramsey for triangles.

Keywords

Random Graph Complete Graph Edge Density 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag 1986

Authors and Affiliations

  • P. Frankl
    • 1
  • V. Rödl
    • 2
    • 3
  1. 1.CNRSQuai Anatole FranceParisFrance
  2. 2.Department of MathematicsFJFI, ČVUTPraha 1Czechoslovakia
  3. 3.AT&T Bell LaboratoriesMurray HillUSA

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