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Combinatorica

, Volume 25, Issue 2, pp 125–141 | Cite as

Sharp Bounds For Some Multicolor Ramsey Numbers

  • Noga Alon*
  • Vojtěch Rödl†
Original Paper

Let H 1,H 2, . . .,H k+1 be a sequence of k+1 finite, undirected, simple graphs. The (multicolored) Ramsey number r(H 1,H 2,...,H k+1) is the minimum integer r such that in every edge-coloring of the complete graph on r vertices by k+1 colors, there is a monochromatic copy of H i in color i for some 1≤ik+1. We describe a general technique that supplies tight lower bounds for several numbers r(H 1,H 2,...,H k+1) when k≥2, and the last graph H k+1 is the complete graph K m on m vertices. This technique enables us to determine the asymptotic behaviour of these numbers, up to a polylogarithmic factor, in various cases. In particular we show that r(K 3,K 3,K m ) = Θ(m 3 poly logm), thus solving (in a strong form) a conjecture of Erdőos and Sós raised in 1979. Another special case of our result implies that r(C 4,C 4,K m ) = Θ(m 2 poly logm) and that r(C 4,C 4,C 4,K m ) = Θ(m 2/log2 m). The proofs combine combinatorial and probabilistic arguments with spectral techniques and certain estimates of character sums.

Mathematics Subject Classification (2000):

05C55 

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Copyright information

© János Bolyai Mathematical Society 2005

Authors and Affiliations

  1. 1.Institute for Advanced StudyPrinceton, NJ 08540USA
  2. 2.Department of MathematicsTel Aviv UniversityTel Aviv 69978Israel
  3. 3.Department of Mathematics and Computer ScienceEmory UniversityAtlantaUSA

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