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The Induced Size-Ramsey Number of Cycles

  • P. E. Haxell (a1), Y. Kohayakawa (a2) and T. Łuczak (a3)

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 C 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 P 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 C 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.

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[1]Alon, N. and Spencer, J. H. (1992) The Probabilistic Method. Wiley, New York.
[2]Beck, J. (1983) On size Ramsey number of paths, trees and circuits I. J. Graph Theory 7, 115129.
[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, 3445.
[4]Beck, J. (1983) An upper bound for diagonal Ramsey numbers. Studia Sci. Math. Hung. 18, 401406.
[5]Bollobás, B. (1985) Random Graphs. Academic Press, London.
[6]Bollobás, B. (1992) Personal communication.
[7]Chen, G. and Schelp, R. H. (1993) Graphs with linearly bounded Ramsey numbers. J. Combinatorial Theory (B) 57, 138149.
[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, 239243.
[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, 323332.
[10]Erdős, P., Faudree, R. J., Rousseau, C. C. and Schelp, R. H. (1978) The size Ramsey number. Periodica Math. Hungar. 9, 145161.
[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, 585595.
[12]Erdős, P. and Rousseau, C. C. (1993) The size Ramsey number of a complete bipartite graph. Discrete Math. 113, 259262.
[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, 111153.
[14]Haxell, P. E. and Kohayakawa, Y. (1995) The size-Ramsey number of trees. Israel J. Math. 89, 261274.
[15]Hoeffding, W. (1963) Probability inequalities for sums of bounded random variables. J. Am. Statistical Assoc. 58, 1330.
[16]Ke, X. (1993) The size Ramsey number of trees with bounded degree. Random Structures and Algorithms 4, 8597.
[17]Kohayakawa, Y. (1993) The regularity lemma of Szeméredi for sparse graphs. Manuscript.
[18]Luczak, T. (1993) The size of the largest hole in a random graph. Discrete Math. 112, 151163.
[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, 148188.
[20]Rödl, V. (1973) The dimension of a graph and generalized Ramsey theorems. Thesis, Charles University, Praha.
[21]Rödl, V. (1993) Personal communication.
[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, 399401.
[23]Turán, P. (1941) On an extremal problem in graph theory. Mat. Fiz. Lapok 48, 436452. (in Hungarian)

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The Induced Size-Ramsey Number of Cycles

  • P. E. Haxell (a1), Y. Kohayakawa (a2) and T. Łuczak (a3)

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