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The Induced Size-Ramsey Number of Cycles

Published online by Cambridge University Press:  12 September 2008

P. E. Haxell ,
Y. Kohayakawa  and
T. Łuczak
P. E. Haxell
Affiliation:
Department of Combinatorics and Optimisation, University of Waterloo, Waterloo, Ontario, Canada0N2L 3G1 (email: [email protected])
Y. Kohayakawa
Affiliation:
Instituto de Matemática e Estatística, Universidade de São Paulo, Rua do Matão 1010, Cidade Universitaria, 05508–900 São Paulo, SP, Brazil (email: [email protected])
T. Łuczak
Affiliation:
Mathematical Institute of the Polish Academy of Sciences, Poznań, Poland, (email: [email protected])¶
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Abstract

For a graph H and an integer r ≥ 2, the induced r-size-Ramsey number of H is defined to be the smallest integer m for which there exists a graph G with m edges with the following property: however one colours the edges of G with r colours, there always exists a monochromatic induced subgraph H′ of G that is isomorphic to H. This is a concept closely related to the classical r-size-Ramsey number of Erdős, Faudree, Rousseau and Schelp, and to the r-induced Ramsey number, a natural notion that appears in problems and conjectures due to, among others, Graham and Rödl, and Trotter. Here, we prove a result that implies that the induced r-size-Ramsey number of the cycle Cℓ is at most crℓ for some constant cr that depends only upon r. Thus we settle a conjecture of Graham and Rödl, which states that the above holds for the path Pℓ of order ℓ and also generalise in part a result of Bollobás, Burr and Reimer that implies that the r-size Ramsey number of the cycle Cℓ is linear in ℓ Our method of proof is heavily based on techniques from the theory of random graphs and on a variant of the powerful regularity lemma of Szemerédi.

Type
Research Article
Information
Combinatorics, Probability and Computing , Volume 4 , Issue 3 , September 1995 , pp. 217 - 239
Copyright
Copyright © Cambridge University Press 1995

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