Skip to main content

We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.

Close cookie message
Cancel
×
×
Home
  • Home
Article
  • Cited by 11
  • Cited by
    Crossref Citations
    Crossref logo
    This article has been cited by the following publications. This list is generated based on data provided by CrossRef.
    Bustamante, Sebastián Hàn, Hiêp and Stein, Maya 2018. Almost partitioning 2-colored complete 3-uniform hypergraphs into two monochromatic tight or loose cycles. Journal of Graph Theory,
    Allen, Peter Böttcher, Julia Cooley, Oliver and Mycroft, Richard 2017. Tight cycles and regular slices in dense hypergraphs. Journal of Combinatorial Theory, Series A, Vol. 149, Issue. , p. 30.
    Maherani, L. and Omidi, G.R. 2017. Monochromatic Hamiltonian Berge-cycles in colored hypergraphs. Discrete Mathematics, Vol. 340, Issue. 8, p. 2043.
    Bustamante, Sebastián Hàn, Hiệp and Stein, Maya 2017. Almost partitioning 2-edge-colourings of 3-uniform hypergraphs with two monochromatic tight cycles. Electronic Notes in Discrete Mathematics, Vol. 61, Issue. , p. 185.
    Morris, Walter and Soltan, Valeriu 2016. Open Problems in Mathematics. p. 351.
    Peng, Xing 2016. The Ramsey number of generalized loose paths in hypergraphs. Discrete Mathematics, Vol. 339, Issue. 2, p. 539.
    Radziszowski, Stanisław P. 2011. Ramsey Theory. p. 41.
    GYÁRFÁS, ANDRÁS and SÁRKÖZY, GÁBOR N. 2011. The 3-Colour Ramsey Number of a 3-Uniform Berge Cycle. Combinatorics, Probability and Computing, Vol. 20, Issue. 01, p. 53.
    Gyárfás, András Sárközy, Gábor N. and Szemerédi, Endre 2010. Long Monochromatic Berge Cycles in Colored 4-Uniform Hypergraphs. Graphs and Combinatorics, Vol. 26, Issue. 1, p. 71.
    Gyárfás, András Sárközy, Gábor N. and Szemerédi, Endre 2010. Monochromatic Matchings in the Shadow Graph of Almost Complete Hypergraphs. Annals of Combinatorics, Vol. 14, Issue. 2, p. 245.
    Cooley, Oliver Fountoulakis, Nikolaos Kühn, Daniela and Osthus, Deryk 2009. Embeddings and Ramsey numbers of sparse κ-uniform hypergraphs. Combinatorica, Vol. 29, Issue. 3, p. 263.
    ×
  • Get access
    Add to cart USD35.00
    Check if you have access via personal or institutional login
    Log in Register

The Ramsey Number for 3-Uniform Tight Hypergraph Cycles

  • P. E. HAXELL (a1), T. ŁUCZAK (a2), Y. PENG (a3), V. RÖDL (a4), A. RUCIŃSKI (a2) and J. SKOKAN (a5)...
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.

Copyright
COPYRIGHT: © Cambridge University Press 2009
References
Hide All
[1]Bondy, J. A. and Erdős, P. (1973) Ramsey numbers for cycles in graphs. J. Combin. Theory Ser. B 14 46–54.
[2]Conlon, D., Fox, J. and Sudakov, B. Ramsey numbers of sparse hypergraphs. Random Struct. Algorithms, to appear.
[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.
[4]Cooley, O., Fountoulakis, N., Kühn, D. and Osthus, D. Embeddings and Ramsey numbers of sparse k-uniform hypergraphs. Combinatorica, to appear.
[5]Faudree, R. and Schelp, R. (1974) All Ramsey numbers for cycles in graphs. Discrete Math. 8 313–329.
[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.
[7]Frankl, P. and Rödl, V. (2002) Extremal problems on set systems. Random Struct. Algorithms 20 131–164.
[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.
[9]Gyárfás, A., Sárközy, G. and Szemerédi, E. Long monochromatic Berge cycles in colored 4-uniform hypergraphs. Submitted.
[10]Gyárfás, A., Sárközy, G. and Szemerédi, E. Monochromatic Hamiltonian 3-tight Berge cycles in 2-colored 4-uniform hypergraphs. Submitted.
[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.
[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.
[13]Łuczak, T. (1999) R(C n, C n, C n) ≤ (4 + o(1))n. J. Combin. Theory Ser. B 75 174–187.
[14]Nagle, B., Olsen, S., Rödl, V. and Schacht, M. (2008) On the Ramsey number of sparse 3-graphs. Graphs Combin. 24 205–228.
[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.
[16]Radziszowski, S. P. (1994) Small Ramsey numbers. Electronic J. Combin. 1, Dynamic Survey 1, 30 pp.
[17]Rödl, V., Ruciński, A. and Szemerédi, E. (2006) A Dirac-type theorem for 3-uniform hypergraphs. Combin. Probab. Comput. 15 253–279.
[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.
[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.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Combinatorics, Probability and Computing
  • ISSN: 0963-5483
  • EISSN: 1469-2163
  • URL: /core/journals/combinatorics-probability-and-computing
Please enter your name
Please enter a valid email address
Who would you like to send this to? *

CAPTCHA *

Skip to the audio challenge
×

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed

Cancel
Confirm
×