Ramsey goodness and beyond

Abstract

In a seminal paper from 1983, Burr and Erdős started the systematic study of Ramsey numbers of cliques vs. large sparse graphs, raising a number of problems. In this paper we develop a new approach to such Ramsey problems using a mix of the Szemerédi regularity lemma, embedding of sparse graphs, Turán type stability, and other structural results. We give exact Ramsey numbers for various classes of graphs, solving five — all but one — of the Burr-Erdős problems.

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References

  1. [1]

    B. Andrásfai, P. Erdős and V. T. Sós: On the connection between chromatic number, maximal clique and minimal degree of a graph, Discrete Math. 8(3) (1974), 205–218.

    MATH  Article  MathSciNet  Google Scholar 

  2. [2]

    B. Bollobás: Modern Graph Theory, Graduate Texts in Mathematics 184, Springer-Verlag, New York (1998), xiv+394 pp.

    Google Scholar 

  3. [3]

    B. Bollobás and V. Nikiforov: Joints in graphs, Discrete Math. 308 (2008), 9–19.

    MATH  Article  MathSciNet  Google Scholar 

  4. [4]

    S. Brandt: Expanding graphs and Ramsey numbers, available at Bielefeld preprint server, (1996), Preprint No. A 96-24.

  5. [5]

    S. A. Burr, P. Erdős, R. J. Faudree, C. C. Rousseau, R. H. Schelp, R. J. Gould and M. S. Jacobson: Goodness of trees for generalized books, Graphs Combin. 3 (1987), 1–6.

    MATH  Article  MathSciNet  Google Scholar 

  6. [6]

    S. A. Burr, R. J. Faudree, C. C. Rousseau and R. H. Schelp: On Ramsey numbers involving starlike multipartite graphs, J. Graph Theory 7 (1983), 395–409.

    MATH  Article  MathSciNet  Google Scholar 

  7. [7]

    S. A. Burr and P. Erdős: Generalizations of a Ramsey-theoretic result of Chvátal, J. Graph Theory 7 (1983), 39–51.

    MATH  Article  MathSciNet  Google Scholar 

  8. [8]

    S. A. Burr: Ramsey numbers involving graphs with long suspended paths, J. London Math. Soc. (2) 24 (1981), 405–413.

    MATH  Article  MathSciNet  Google Scholar 

  9. [9]

    S. A. Burr: Multicolor Ramsey numbers involving graphs with long suspended paths, Discrete Math. 40 (1982), 11–20.

    MATH  Article  MathSciNet  Google Scholar 

  10. [10]

    S. A. Burr: What we can hope to accomplish in generalized Ramsey theory, Discrete Math. 67 (1987), 215–225.

    MATH  Article  MathSciNet  Google Scholar 

  11. [11]

    P. Erdős, R. J. Faudree, C. C. Rousseau and R. H. Schelp: Multipartite graph — sparse graph Ramsey numbers, Combinatorica 5(4) (1985), 311–318.

    Article  MathSciNet  Google Scholar 

  12. [12]

    P. Erdős, R. J. Faudree, C. C. Rousseau and R. H. Schelp: The book-tree Ramsey numbers, Scientia, Series A: Mathematical Sciences 1 (1988), 111–117.

    MathSciNet  Google Scholar 

  13. [13]

    R. J. Faudree, C. C. Rousseau and J. Sheehan: More from the good book, Congress. Numer. XXI, Utilitas Math., Winnipeg, Man., 1978, pp. 289–299.

    Google Scholar 

  14. [14]

    R. J. Faudree, C. C. Rousseau and J. Sheehan: Strongly regular graphs and finite Ramsey theory, Linear Algebra Appl. 46 (1982), 221–241.

    MATH  Article  MathSciNet  Google Scholar 

  15. [15]

    R. J. Faudree, C. C. Rousseau and R. H. Schelp: A good idea in Ramsey theory, in: Graph theory, combinatorics, algorithms, and applications (San Francisco, CA, 1989), pp. 180–189, SIAM, Philadelphia, PA, 1991.

    Google Scholar 

  16. [16]

    R. J. Faudree, C. C. Rousseau and J. Sheehan: Cycle-book Ramsey numbers, Ars Combin. 31 (1991), 239–248.

    MATH  MathSciNet  Google Scholar 

  17. [17]

    R. J. Faudree, R. H. Schelp and C. C. Rousseau: Generalizations of a Ramsey result of Chvátal, in: The theory and applications of graphs (Kalamazoo, Mich., 1980), pp. 351–361, Wiley, New York, 1981.

    Google Scholar 

  18. [18]

    J. Komlós and M. Simonovits: Szemerédi’s regularity lemma and its applications in graph theory, in: Combinatorics, Paul Erdős is Eighty, Vol. 2 (Keszthely, 1993), Bolyai Soc. Math. Stud., 2, János Bolyai Math. Soc., Budapest, 1996, pp. 295–352.

    Google Scholar 

  19. [19]

    A. Kostochka and V. Rödl: On graphs with small Ramsey numbers, J. Graph Theory 37 (2001), 109–204.

    Article  Google Scholar 

  20. [20]

    J. Komlós, G. N. Sárközy and E. Szemerédi: Blow-up lemma, Combinatorica 17(1) (1997), 109–123.

    MATH  Article  MathSciNet  Google Scholar 

  21. [21]

    Y. Li and C. C. Rousseau: Fan-complete graph Ramsey numbers, J. Graph Theory 23 (1996), 413–420.

    MATH  Article  MathSciNet  Google Scholar 

  22. [22]

    V. Nikiforov: Edge distribution of graphs with few induced copies of a given graph, Combin. Probab. Comput. 15 (2006), 895–902.

    MATH  Article  MathSciNet  Google Scholar 

  23. [23]

    V. Nikiforov: Graphs with many r-cliques have large complete r-partite subgraphs, Bull. London Math. Soc. 40(1) (2008), 23–25.

    MATH  Article  MathSciNet  Google Scholar 

  24. [24]

    V. Nikiforov and C. C. Rousseau: Large generalized books are p-good, J. Combin. Theory Ser. B 92(1) (2004), 85–97.

    MATH  Article  MathSciNet  Google Scholar 

  25. [25]

    V. Nikiforov and C. C. Rousseau: A note on Ramsey numbers for books, J. Graph Theory 49 (2005), 168–176.

    MATH  Article  MathSciNet  Google Scholar 

  26. [26]

    V. Nikiforov and C. C. Rousseau: Book Ramsey numbers I, Random Structures Algorithms 27 (2005), 379–400.

    MATH  Article  MathSciNet  Google Scholar 

  27. [27]

    V. Nikiforov and C. C. Rousseau: Ramsey Goodness and Beyond, preprint available at arXiv:math/0703653.

  28. [28]

    C. C. Rousseau and J. Sheehan: On Ramsey numbers for books, J. Graph Theory 2 (1978), 77–87.

    MATH  Article  MathSciNet  Google Scholar 

  29. [29]

    C. C. Rousseau and J. Sheehan: A class of Ramsey problems involving trees, J. London Math. Soc. (2) 18 (1978), 392–396.

    MATH  Article  MathSciNet  Google Scholar 

  30. [30]

    B. Sudakov: Large K r-free subgraphs in K s-free graphs and some other Ramsey-type problems, Random Structures Algorithms 26 (2005), 253–265.

    MATH  Article  MathSciNet  Google Scholar 

  31. [31]

    E. Szemerédi: Regular partitions of graphs, in: Problèmes combinatoires et théorie des graphes (Colloq. Internat. CNRS, Univ. Orsay, Orsay, 1976), pp. 399–401, 260, CNRS, Paris, 1978.

    Google Scholar 

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Nikiforov, V., Rousseau, C.C. Ramsey goodness and beyond. Combinatorica 29, 227–262 (2009). https://doi.org/10.1007/s00493-009-2409-2

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Mathematics Subject Classification (2000)

  • 05C55
  • 05C35