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    This article has been cited by the following publications. This list is generated based on data provided by CrossRef.

    Morris, Walter and Soltan, Valeriu 2016. Open Problems in Mathematics.


    Peng, Xing 2016. The Ramsey number of generalized loose paths in hypergraphs. Discrete Mathematics, Vol. 339, Issue. 2, p. 539.


    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. Monochromatic Matchings in the Shadow Graph of Almost Complete Hypergraphs. Annals of Combinatorics, Vol. 14, Issue. 2, p. 245.


    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.


    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.


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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)
  • DOI: http://dx.doi.org/10.1017/S096354830800967X
  • Published online: 01 March 2009
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.

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[1]J. A. Bondy and P. Erdős (1973) Ramsey numbers for cycles in graphs. J. Combin. Theory Ser. B 14 4654.

[3]O. Cooley , N. Fountoulakis , D. Kühn and D. Osthus (2008) 3-uniform hypergraphs of bounded degree have linear Ramsey numbers. J. Combin. Theory Ser. B 98 484505.

[5]R. Faudree and R. Schelp (1974) All Ramsey numbers for cycles in graphs. Discrete Math. 8 313329.

[6]A. Figaj and T. Łuczak (2007) The Ramsey number for a triple of long even cycles. J. Combin. Theory Ser. B 97 584596.

[7]P. Frankl and V. Rödl (2002) Extremal problems on set systems. Random Struct. Algorithms 20 131164.

[11]P. Haxell , T. Łuczak , Y. Peng , V. Rödl , A. Ruciński , M. Simonovits and J. Skokan (2006) The Ramsey number for hypergraph cycles I. J. Combin. Theory Ser. A 113 6783.

[13]T. Łuczak (1999) R(Cn, Cn, Cn) ≤ (4 + o(1))n. J. Combin. Theory Ser. B 75 174187.

[14]B. Nagle , S. Olsen , V. Rödl and M. Schacht (2008) On the Ramsey number of sparse 3-graphs. Graphs Combin. 24 205228.

[15]J. Polcyn , V. Rödl , A. Ruciński and E. Szemerédi (2006) Short paths in quasi-random triple systems with sparse underlying graphs. J. Combin. Theory Ser. B 96 584607.

[17]V. Rödl , A. Ruciński and E. Szemerédi (2006) A Dirac-type theorem for 3-uniform hypergraphs. Combin. Probab. Comput. 15 253279.

[18]V. Rosta (1973) On a Ramsey-type problem of J. A. Bondy and P. Erdős, I and II. J. Combin. Theory Ser. B 15 94104, 105–120.

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Combinatorics, Probability and Computing
  • ISSN: 0963-5483
  • EISSN: 1469-2163
  • URL: /core/journals/combinatorics-probability-and-computing
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