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New Upper Bound on Vertex Folkman Numbers

  • Andrzej Dudek
  • Vojtěch Rödl
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4957)

Abstract

In 1970, J. Folkman proved that for a given integer r and a graph G of order n there exists a graph H with the same clique number as G such that every r coloring of vertices of H yields at least one monochromatic copy of G. His proof gives no good bound on the order of graph H, i.e. the order of H is bounded by an iterated power function. A related problem was studied by Łuczak, Ruciński and Urbański, who gave some explicite bound on the order of H when G is a clique. In this note we give an alternative proof of Folkman’s theorem with the relatively small order of H bounded from above by O(n 3log3 n). This improves Łuczak, Ruciński and Urbański’s result.

Keywords

Ramsey theory vertex Folkman numbers 

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Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Andrzej Dudek
    • 1
  • Vojtěch Rödl
    • 1
  1. 1.Department of Mathematics and Computer ScienceEmory UniversityAtlantaUSA

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