Hostname: page-component-586b7cd67f-t8hqh Total loading time: 0 Render date: 2024-11-25T00:43:31.799Z Has data issue: false hasContentIssue false
  • English
  • Français

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])¶
Get access Rights & Permissions [Opens in a new window]

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Alon, N. and Spencer, J. H. (1992) The Probabilistic Method. Wiley, New York.Google Scholar
[2]Beck, J. (1983) On size Ramsey number of paths, trees and circuits I. J. Graph Theory 7, 115–129.CrossRefGoogle Scholar
[3]Beck, J. (1990) On size Ramsey number of paths, trees and circuits II. Mathematics of Ramsey Theory (Nešetřil, J. and Rödl, V., eds.), Springer-Verlag, Berlin, 34–45.CrossRefGoogle Scholar
[4]Beck, J. (1983) An upper bound for diagonal Ramsey numbers. Studia Sci. Math. Hung. 18, 401–406.Google Scholar
[5]Bollobás, B. (1985) Random Graphs. Academic Press, London.Google Scholar
[6]Bollobás, B. (1992) Personal communication.Google Scholar
[7]Chen, G. and Schelp, R. H. (1993) Graphs with linearly bounded Ramsey numbers. J. Combinatorial Theory (B) 57, 138–149.CrossRefGoogle Scholar
[8]Chvátal, V., Rödl, V., Szemerédi, E. and Trotter, W. T. (1983) The Ramsey number of a graph with bounded degree. J. Combinatorial Theory (B) 34, 239–243.CrossRefGoogle Scholar
[9]Deuber, W. (1975) A generalization of Ramsey's theorem. Infinite and Finite Sets, Colloq. Math. Soc. J. Bolyai 10 (Hajnal, A., Rado, R. and Sós, V. T., eds.), North Holland, Amsterdam, 323–332.Google Scholar
[10]Erdős, P., Faudree, R. J., Rousseau, C. C. and Schelp, R. H. (1978) The size Ramsey number. Periodica Math. Hungar. 9, 145–161.CrossRefGoogle Scholar
[11]Erdős, P., Hajnal, A. and Pósa, L. (1975) Strong embeddings of graphs into colored graphs. Infinite and Finite Sets, Colloq. Math. Soc. J. Bolyai 10, (Hajnal, A., Rado, R. and Sós, V. T., eds.), North Holland, Amsterdam, 585–595.Google Scholar
[12]Erdős, P. and Rousseau, C. C. (1993) The size Ramsey number of a complete bipartite graph. Discrete Math. 113, 259–262.CrossRefGoogle Scholar
[13]Graham, R. L. and Rödl, V. (1987) Numbers in Ramsey theory. Surveys in Combinatorics, London Mathematical Society Lecture Note Series 123 (Whitehead, C., ed.), Cambridge University Press, Cambridge, 111–153.Google Scholar
[14]Haxell, P. E. and Kohayakawa, Y. (1995) The size-Ramsey number of trees. Israel J. Math. 89, 261–274.CrossRefGoogle Scholar
[15]Hoeffding, W. (1963) Probability inequalities for sums of bounded random variables. J. Am. Statistical Assoc. 58, 13–30.CrossRefGoogle Scholar
[16]Ke, X. (1993) The size Ramsey number of trees with bounded degree. Random Structures and Algorithms 4, 85–97.CrossRefGoogle Scholar
[17]Kohayakawa, Y. (1993) The regularity lemma of Szeméredi for sparse graphs. Manuscript.Google Scholar
[18]Luczak, T. (1993) The size of the largest hole in a random graph. Discrete Math. 112, 151–163.CrossRefGoogle Scholar
[19]McDiarmid, C. J. H. (1989) On the method of bounded differences. Surveys in Combinatorics 1989, London Mathematical Society Lecture Notes Series 141 (Siemons, J., ed.), Cambridge University Press, Cambridge, 148–188.Google Scholar
[20]Rödl, V. (1973) The dimension of a graph and generalized Ramsey theorems. Thesis, Charles University, Praha.Google Scholar
[21]Rödl, V. (1993) Personal communication.Google Scholar
[22]Szemerédi, E. (1978) Regular partitions of graphs. Problèmes en Combinatoire et Théorie des Graphes, Proc. Colloque Inter. CNRS (Bermond, J.-C., Fournier, J.-C., Las Vergnas, M. and Sotteau, D., eds.), CNRS, Paris, 399–401.Google Scholar
[23]Turán, P. (1941) On an extremal problem in graph theory. Mat. Fiz. Lapok 48, 436–452. (in Hungarian)Google Scholar