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Ramsey goodness and beyond

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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. 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.

    Article  MATH  MathSciNet  Google Scholar 

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

    MATH  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

  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.

    Article  MATH  MathSciNet  Google Scholar 

  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.

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

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

    Article  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  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.

    Article  MATH  MathSciNet  Google Scholar 

  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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Authors and Affiliations

  1. Department of Mathematical Sciences, University of Memphis, Memphis, TN, 38152, USA

    Vladimir Nikiforov & Cecil C. Rousseau

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  1. Vladimir Nikiforov
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  2. Cecil C. Rousseau
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Correspondence to Vladimir Nikiforov.

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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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  • DOI: https://doi.org/10.1007/s00493-009-2409-2

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