Advertisement

Combinatorica

, Volume 29, Issue 3, pp 263–297 | Cite as

Embeddings and Ramsey numbers of sparse κ-uniform hypergraphs

  • Oliver Cooley
  • Nikolaos Fountoulakis
  • Daniela Kühn
  • Deryk Osthus
Article

Abstract

Chvátal, Rödl, Szemerédi and Trotter [3] proved that the Ramsey numbers of graphs of bounded maximum degree are linear in their order. In [6,23] the same result was proved for 3-uniform hypergraphs. Here we extend this result to κ-uniform hypergraphs for any integer κ ≥ 3. As in the 3-uniform case, the main new tool which we prove and use is an embedding lemma for κ-uniform hypergraphs of bounded maximum degree into suitable κ-uniform ‘quasi-random’ hypergraphs.

Mathematics Subject Classification (2000)

05D10 05C65 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    G. Chen and R. Schelp: Graphs with linearly bounded Ramsey numbers, J. Combinatorial Theory B 57 (1993), 138–149.zbMATHCrossRefMathSciNetGoogle Scholar
  2. [2]
    S. A. Burr and P. Erdős: On the magnitude of generalized Ramsey numbers for graphs; in: Infinite and Finite Sets I., Colloquia Mathematica Societatis János Bolyai vol. 10 (1975), 214–240.Google Scholar
  3. [3]
    V. Chvátal, V. Rödl, E. Szemerédi and W. T. Trotter, Jr.: The Ramsey number of a graph with a bounded maximum degree, J. Combinatorial Theory B 34 (1983), 239–243.zbMATHCrossRefGoogle Scholar
  4. [4]
    D. Conlon, J. Fox and B. Sudakov: Ramsey numbers of sparse hypergraphs, Random Structures & Algorithms, to appear.Google Scholar
  5. [5]
    O. Cooley: Ph.D. thesis, University of Birmingham, in preparation.Google Scholar
  6. [6]
    O. Cooley, N. Fountoulakis, D. Kühn and D. Osthus: 3-uniform hypergraphs of bounded degree have linear Ramsey numbers, J. Combinatorial Theory B 98 (2008), 484–505.zbMATHCrossRefGoogle Scholar
  7. [7]
    P. Erdős and R. Rado: Combinatorial theorems on classifications of subsets of a given set, Proc. London Mathematical Society 3 (1952), 417–439.CrossRefGoogle Scholar
  8. [8]
    P. Frankl and V. Rödl: Extremal problems on set systems, Random Structures & Algorithms 20 (2002), 131–164.zbMATHCrossRefMathSciNetGoogle Scholar
  9. [9]
    W. T. Gowers: Hypergraph regularity and the multidimensional Szemerédi theorem, Ann. of Math. 166 (2007), 897–946.zbMATHCrossRefMathSciNetGoogle Scholar
  10. [10]
    W. T. Gowers: Quasirandomness, counting and regularity for 3-uniform hypergraphs; Combinatorics, Probability & Computing 15 (2006), 143–184.zbMATHCrossRefMathSciNetGoogle Scholar
  11. [11]
    R. L. Graham, B. L. Rothschild and J. H. Spencer: Ramsey Theory, John Wiley & Sons, 1980.Google Scholar
  12. [12]
    R. L. Graham, V. Rödl and A. Ruciński: On graphs with linear Ramsey numbers, J. Graph Theory 35 (2000), 176–192.zbMATHCrossRefMathSciNetGoogle Scholar
  13. [13]
    A. Gyárfás, J. Lehel, G. N. Sárközy and R. Schelp: Monochromatic Hamiltonian Berge-cycles in colored complete uniform hypergraphs, J. Combinatorial Theory B 98 (2008), 342–358.zbMATHCrossRefGoogle Scholar
  14. [14]
    P. E. Haxell, T. Łuczak, Y. Peng, V. Rödl, A. Ruciński, M. Simonovits and J. Skokan: The Ramsey number for hypergraph cycles I, J. Combinatorial Theory A 113 (2006), 67–83.zbMATHCrossRefGoogle Scholar
  15. [15]
    P. E. Haxell, T. Łuczak, Y. Peng, V. Rödl, A. Ruciński and J. Skokan: The Ramsey number for 3-uniform tight hypergraph cycles, Combinatorics, Probability & Computing 18 (2009), 165–203.CrossRefGoogle Scholar
  16. [16]
    Y. Ishigami: Linear Ramsey numbers for bounded-degree hypergraphs, preprint.Google Scholar
  17. [17]
    P. Keevash: A hypergraph blowup lemma, preprint.Google Scholar
  18. [18]
    Y. Kohayakawa, V. Rödl and J. Skokan: Hypergraphs, quasi-randomness, and conditions for regularity; J. Combinatorial Theory A 97 (2002), 307–352.zbMATHCrossRefGoogle Scholar
  19. [19]
    J. Komlós, G. Sárkőzy and E. Szemerédi: The blow-up lemma, Combinatorica 17 (1997), 109–123.zbMATHCrossRefMathSciNetGoogle Scholar
  20. [20]
    J. Komlós and M. Simonovits: Szemerédi’s Regularity Lemma and its applications in graph theory, Bolyai Society Mathematical Studies 2, Combinatorics, Paul Erdós is Eighty, (Vol. 2) (D. Miklós, V. T. Sós and T. Szőnyi eds.), Budapest (1996), 295–352.Google Scholar
  21. [21]
    A. Kostochka and V. Rödl: On Ramsey numbers of uniform hypergraphs with given maximum degree, J. Combinatorial Theory A 113 (2006), 1555–1564.zbMATHCrossRefGoogle Scholar
  22. [22]
    D. Kühn and D. Osthus: Loose Hamilton cycles in 3-uniform hypergraphs of large minimum degree, J. Combinatorial Theory B 96 (2006), 767–821.zbMATHCrossRefGoogle Scholar
  23. [23]
    B. Nagle, S. Olsen, V. Rödl and M. Schacht: On the Ramsey number of sparse 3-graphs, Graphs and Combinatorics 24 (2008), 205–228.zbMATHCrossRefMathSciNetGoogle Scholar
  24. [24]
    B. Nagle and V. Rödl: Regularity properties for triple systems, Random Structures & Algorithms 23 (2003), 264–332.zbMATHCrossRefMathSciNetGoogle Scholar
  25. [25]
    B. Nagle, V. Rödl and M. Schacht: The counting lemma for κ-uniform hypergraphs, Random Structures & Algorithms 28 (2006), 113–179.zbMATHCrossRefMathSciNetGoogle Scholar
  26. [26]
    J. Polcyn, V. Rödl, A. Ruciński and E. Szemerédi: Short paths in quasi-random triple systems with sparse underlying graphs, J. Combinatorial Theory B 96 (2006), 584–607.zbMATHCrossRefGoogle Scholar
  27. [27]
    V. Rödl and M. Schacht: Regular partitions of hypergraphs: Regularity Lemma; Combinatorics, Probability & Computing 16 (2007), 833–885.zbMATHGoogle Scholar
  28. [28]
    V. Rödl and M. Schacht: Regular partitions of hypergraphs: Counting Lemmas; Combinatorics, Probability & Computing 16 (2007), 887–901.zbMATHGoogle Scholar
  29. [29]
    V. Rödl and J. Skokan: Regularity lemma for κ-uniform hypergraphs, Random Structures & Algorithms 25 (2004), 1–42.zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© János Bolyai Mathematical Society and Springer Verlag 2009

Authors and Affiliations

  • Oliver Cooley
    • 1
  • Nikolaos Fountoulakis
    • 1
  • Daniela Kühn
    • 1
  • Deryk Osthus
    • 1
  1. 1.School of MathematicsUniversity of BirminghamEdgbastonUK

Personalised recommendations