On the Ramsey multiplicity of complete graphs

Abstract

We show that, for n large, there must exist at least

$$\frac{{n^t }} {{C^{(1 + o(1)t^2 } )}}$$

monochromatic K t s in any two-colouring of the edges of K n , where C≈2.18 is an explicitly defined constant. The old lower bound, due to Erdős [2], and based upon the standard bounds for Ramsey’s theorem, is

$$\frac{{n^t }} {{4^{(1 + o(1)t^2 } )}}. $$

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Correspondence to David Conlon.

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The author is supported by a research fellowship at St John’s College, Cambridge, but was also supported for part of the time that this work was being carried out by the MRTN-CT-2004-511953 project at the Alfréd Rényi Institute of Mathematics in Budapest.

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Conlon, D. On the Ramsey multiplicity of complete graphs. Combinatorica 32, 171–186 (2012). https://doi.org/10.1007/s00493-012-2465-x

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Mathematics Subject Classification (2010)

  • 05C55
  • 05D10