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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)...

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.

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[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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