Hostname: page-component-6bf8c574d5-gr6zb Total loading time: 0 Render date: 2025-02-17T23:03:20.733Z Has data issue: false hasContentIssue false
  • English
  • Français

The Ramsey Number for 3-Uniform Tight Hypergraph Cycles

Published online by Cambridge University Press:  01 March 2009

P. E. HAXELL ,
T. ŁUCZAK ,
Y. PENG ,
V. RÖDL ,
A. RUCIŃSKI  and
J. SKOKAN
P. E. HAXELL
Affiliation:
Department of Combinatorics and Optimization, University of Waterloo, Waterloo, Ontario, CanadaN2L 3G1 (e-mail: [email protected])
T. ŁUCZAK
Affiliation:
Department of Discrete Mathematics, Adam Mickiewicz University, 61-614 Poznań, Poland (e-mail: [email protected], [email protected])
Y. PENG
Affiliation:
Department of Mathematics and Computer Science, Indiana State University, Terre Haute, IN 47809, USA (e-mail: [email protected])
V. RÖDL
Affiliation:
Department of Mathematics and Computer Science, Emory University, Atlanta, GA 30032, USA (e-mail: [email protected])
A. RUCIŃSKI
Affiliation:
Department of Discrete Mathematics, Adam Mickiewicz University, 61-614 Poznań, Poland (e-mail: [email protected], [email protected])
J. SKOKAN
Affiliation:
Department of Mathematics, London School of Economics and Political Science, Houghton Street, London WC2A 2AE, UK (e-mail: [email protected])
Get access Rights & Permissions [Opens in a new window]

Abstract

Let C(3)n denote the 3-uniform tight cycle, that is, the hypergraph with vertices v1, .–.–., vn and edges v1v2v3, v2v3v4, .–.–., vn−1vnv1, vnv1v2. We prove that the smallest integer N = N(n) for which every red–blue colouring of the edges of the complete 3-uniform hypergraph with N vertices contains a monochromatic copy of C(3)n is asymptotically equal to 4n/3 if n is divisible by 3, and 2n otherwise. The proof uses the regularity lemma for hypergraphs of Frankl and Rödl.

Type
Paper
Information
Combinatorics, Probability and Computing , Volume 18 , Issue 1-2: Papers from The 2006 Oberwolfach Meeting on Combinatorics, Probability and Computing , March 2009 , pp. 165 - 203
Copyright
Copyright © Cambridge University Press 2009

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]Bondy, J. A. and Erdős, P. (1973) Ramsey numbers for cycles in graphs. J. Combin. Theory Ser. B 14 46–54.CrossRefGoogle Scholar
[2]Conlon, D., Fox, J. and Sudakov, B. Ramsey numbers of sparse hypergraphs. Random Struct. Algorithms, to appear.Google Scholar
[3]Cooley, O., Fountoulakis, N., Kühn, D. and Osthus, D. (2008) 3-uniform hypergraphs of bounded degree have linear Ramsey numbers. J. Combin. Theory Ser. B 98 484–505.CrossRefGoogle Scholar
[4]Cooley, O., Fountoulakis, N., Kühn, D. and Osthus, D. Embeddings and Ramsey numbers of sparse k-uniform hypergraphs. Combinatorica, to appear.Google Scholar
[5]Faudree, R. and Schelp, R. (1974) All Ramsey numbers for cycles in graphs. Discrete Math. 8 313–329.CrossRefGoogle Scholar
[6]Figaj, A. and Łuczak, T. (2007) The Ramsey number for a triple of long even cycles. J. Combin. Theory Ser. B 97 584–596.CrossRefGoogle Scholar
[7]Frankl, P. and Rödl, V. (2002) Extremal problems on set systems. Random Struct. Algorithms 20 131–164.CrossRefGoogle Scholar
[8]Gyárfás, A., Sárközy, G. and Szemerédi, E. (2008) The Ramsey number of diamond-matchings and loose cycles in hypergraphs. Electron. J. Combin. 15 #R126.CrossRefGoogle Scholar
[9]Gyárfás, A., Sárközy, G. and Szemerédi, E. Long monochromatic Berge cycles in colored 4-uniform hypergraphs. Submitted.Google Scholar
[10]Gyárfás, A., Sárközy, G. and Szemerédi, E. Monochromatic Hamiltonian 3-tight Berge cycles in 2-colored 4-uniform hypergraphs. Submitted.Google Scholar
[11]Haxell, P., Łuczak, T., Peng, Y., Rödl, V., Ruciński, A., Simonovits, M. and Skokan, J. (2006) The Ramsey number for hypergraph cycles I. J. Combin. Theory Ser. A 113 67–83.CrossRefGoogle Scholar
[12]Haxell, P., Łuczak, T., Peng, Y., Rödl, V., Ruciński, A. and Skokan, J. (2007) The Ramsey number for hypergraph cycles II. CDAM Research Report LSE-CDAM-2007-04.Google Scholar
[13]Łuczak, T. (1999) R(C n, C n, C n) ≤ (4 + o(1))n. J. Combin. Theory Ser. B 75 174–187.CrossRefGoogle Scholar
[14]Nagle, B., Olsen, S., Rödl, V. and Schacht, M. (2008) On the Ramsey number of sparse 3-graphs. Graphs Combin. 24 205–228.CrossRefGoogle Scholar
[15]Polcyn, J., Rödl, V., Ruciński, A. and Szemerédi, E. (2006) Short paths in quasi-random triple systems with sparse underlying graphs. J. Combin. Theory Ser. B 96 584–607.CrossRefGoogle Scholar
[16]Radziszowski, S. P. (1994) Small Ramsey numbers. Electronic J. Combin. 1, Dynamic Survey 1, 30 pp.Google Scholar
[17]Rödl, V., Ruciński, A. and Szemerédi, E. (2006) A Dirac-type theorem for 3-uniform hypergraphs. Combin. Probab. Comput. 15 253–279.CrossRefGoogle Scholar
[18]Rosta, V. (1973) On a Ramsey-type problem of J. A. Bondy and P. Erdős, I and II. J. Combin. Theory Ser. B 15 94–104, 105–120.CrossRefGoogle Scholar
[19]Szemerédi, E. (1978) Regular partitions of graphs. In Problèmes Combinatoires et Théorie des Graphes (Colloq. Internat. CNRS, Univ. Orsay, Orsay, 1976), Vol. 260 of Colloq. Internat. CNRS, CNRS, Paris, pp. 399–401.Google Scholar