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Combinatorica

, 31:39 | Cite as

On K s -free subgraphs in K s+k -free graphs and vertex Folkman numbers

  • Andrzej Dudek
  • Vojtěch Rödl
Article

Abstract

Extending the problem of determining Ramsey numbers Erdős and Rogers introduced the following function. For given integers 2 ≤ s < t let f s,t (n) = min{max{|S|: SV (H) and H[S] contains no K s }}, where the minimum is taken over all K t -free graphs H of order n. This function attracted a considerable amount of attention but despite that, the gap between the lower and upper bounds is still fairly wide. For example, when t=s+1, the best bounds have been of the form Ω(n 1/2+o(1)) ≤ f s,s+1(n) ≤ O(n 1−ɛ(s)), where ɛ(s) tends to zero as s tends to infinity. In this paper we improve the upper bound by showing that f s,s+1(n) ≤ O(n 2/3). Moreover, we show that for every ɛ > 0 and sufficiently large integers 1 ≪ ks, Ω(n 1/2−ɛ ) ≤ f s,s+k (n) ≤ O(n 1/2+ɛ . In addition, we also discuss some connections between the function f s,t and vertex Folkman numbers.

Mathematics Subject Classification (2000)

05C35 05C55 

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Copyright information

© János Bolyai Mathematical Society and Springer Verlag 2011

Authors and Affiliations

  1. 1.Department of Mathematical SciencesCarnegie Mellon UniversityPittsburghUSA
  2. 2.Department of Mathematics and Computer ScienceEmory UniversityAtlantaUSA

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