Location via proxy:   [ UP ]  
[Report a bug]   [Manage cookies]                
Skip to main content
Springer Nature Link
Log in

On two problems in graph Ramsey theory

  • Published:
Combinatorica Aims and scope Submit manuscript

Abstract

We study two classical problems in graph Ramsey theory, that of determining the Ramsey number of bounded-degree graphs and that of estimating the induced Ramsey number for a graph with a given number of vertices.

The Ramsey number r(H) of a graph H is the least positive integer N such that every two-coloring of the edges of the complete graph K N contains a monochromatic copy of H. A famous result of Chváatal, Rödl, Szemerédi and Trotter states that there exists a constant c(Δ) such that r(H) ≤ c(Δ)n for every graph H with n vertices and maximum degree Δ. The important open question is to determine the constant c(Δ). The best results, both due to Graham, Rödl and Ruciński, state that there are positive constants c and c′ such that \(2^{c'\Delta } \leqslant c(\Delta ) \leqslant ^{c\Delta \log ^2 \Delta }\). We improve this upper bound, showing that there is a constant c for which c(Δ) ≤ 2cΔlogΔ.

The induced Ramsey number r ind (H) of a graph H is the least positive integer N for which there exists a graph G on N vertices such that every two-coloring of the edges of G contains an induced monochromatic copy of H. Erdős conjectured the existence of a constant c such that, for any graph H on n vertices, r ind (H) ≤ 2cnlogn. We move a step closer to proving this conjecture, showing that r ind (H) ≤ 2cnlogn. This improves upon an earlier result of Kohayakawa, Prömel and Rödl by a factor of logn in the exponent.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. N. Alon, M. Krivelevich and B. Sudakov: Turán numbers of bipartite graphs and related Ramsey-type questions, Combin. Probab. Comput. 12 (2003), 477–494.

    Article  MathSciNet  MATH  Google Scholar 

  2. S. A. Burr and P. Erdős: On the magnitude of generalized Ramsey numbers for graphs, in: Infinite and Finite Sets, Vol. 1 (Keszthely, 1973), Colloq. Math. Soc. János Bolyai, Vol. 10, North-Holland, Amsterdam/London, 1975, 215–240.

    Google Scholar 

  3. G. Chen and R. Schelp, Graphs with linearly bounded Ramsey numbers, J. Combin. Theory Ser. B 57 (1993), 138–149.

    Article  MathSciNet  MATH  Google Scholar 

  4. V. Chvátal, V. Rödl, E. Szemerédi and W.T. Trotter Jr.: The Ramsey number of a graph with bounded maximum degree, J. Combin. Theory Ser. B 34 (1983), 239–243.

    Article  MathSciNet  MATH  Google Scholar 

  5. D. Conlon: A new upper bound for diagonal Ramsey numbers, Annals of Math. 170 (2009), 941–960.

    Article  MathSciNet  MATH  Google Scholar 

  6. D. Conlon: Hypergraph packing and sparse bipartite Ramsey numbers, Combin. Probab. Comput., 18 (2009), 913–923.

    Article  MathSciNet  MATH  Google Scholar 

  7. D. Conlon, J. Fox and B. Sudakov: Ramsey numbers of sparse hypergraphs, Random Structures Algorithms 35 (2009), 1–14.

    Article  MathSciNet  MATH  Google Scholar 

  8. O. Cooley, N. Fountoulakis, D. Kühn and D. Osthus: 3-uniform hypergraphs of bounded degree have linear Ramsey numbers, J. Combin. Theory Ser. B 98 (2008), 484–505.

    Article  MathSciNet  MATH  Google Scholar 

  9. O. Cooley, N. Fountoulakis, D. Kühn and D. Osthus: Embeddings and Ramsey numbers of sparse k-uniform hypergraphs, Combinatorica 28 (2009), 263–297.

    Article  Google Scholar 

  10. W. Deuber: A generalization of Ramsey’s theorem, in: Infinite and Finite Sets, Vol. 1 (Keszthely, 1973), Colloq. Math. Soc. János Bolyai, Vol. 10, North-Holland, Amsterdam/London, 1975, 323–332.

    Google Scholar 

  11. N. Eaton: Ramsey numbers for sparse graphs, Discrete Math. 185 (1998), 63–75.

    Article  MathSciNet  MATH  Google Scholar 

  12. P. Erdős: On some problems in graph theory, combinatorial analysis and combinatorial number theory, in: Graph theory and combinatorics (Cambridge, 1983), Academic Press, London, New York, 1984, 1–17.

    Google Scholar 

  13. P. Erdős: Problems and results on finite and infinite graphs, in: Recent advances in graph theory (Proc. Second Czechoslovak Sympos., Prague, 1974), Academia, Prague, 1975, 183–192.

    Google Scholar 

  14. P. Erdős: Some remarks on the theory of graphs, Bull. Amer. Math. Soc. 53 (1947), 292–294.

    Article  MathSciNet  Google Scholar 

  15. P. Erdős and R. Graham: On partition theorems for finite graphs, in: Infinite and Finite Sets, Vol. 1 (Keszthely, 1973), Colloq. Math. Soc. János Bolyai, Vol. 10, North-Holland, Amsterdam/London, 1975, 515–527.

    Google Scholar 

  16. P. Erdős, A. Hajnal and L. Pósa: Strong embeddings of graphs into colored graphs, in: Infinite and Finite Sets, Vol. 1 (Keszthely, 1973), Colloq. Math. Soc. János Bolyai, Vol. 10, North-Holland, Amsterdam/London, 1975, 585–595.

    Google Scholar 

  17. P. Erdős and G. Szekeres: A combinatorial problem in geometry, Compositio Math. 2 (1935), 463–470.

    MathSciNet  Google Scholar 

  18. J. Fox and B. Sudakov, Induced Ramsey-type theorems, Adv. Math. 219 (2008), 1771–1800.

    Article  MathSciNet  MATH  Google Scholar 

  19. J. Fox and B. Sudakov: Density theorems for bipartite graphs and related Ramsey-type results, Combinatorica 29 (2009), 153–196.

    MathSciNet  MATH  Google Scholar 

  20. J. Fox and B. Sudakov: Two remarks on the Burr-Erdős conjecture, European J. Combin. 30 (2009), 1630–1645.

    Article  MathSciNet  MATH  Google Scholar 

  21. W. T. Gowers: Lower bounds of tower type for Szemerédi’s uniformity lemma, Geom. Funct. Anal. 7 (1997), 322–337.

    Article  MathSciNet  MATH  Google Scholar 

  22. R. L. Graham, V. Rödl and A. Ruciński: On graphs with linear Ramsey numbers, J. Graph Theory 35 (2000), 176–192.

    Article  MathSciNet  MATH  Google Scholar 

  23. R. L. Graham, V. Rödl and A. Ruciński: On bipartite graphs with linear Ramsey numbers, Combinatorica 21 (2001), 199–209.

    Article  MathSciNet  MATH  Google Scholar 

  24. H. A. Kierstead and W. T. Trotter Jr.: Planar graph colorings with an uncooperative partner, J. Graph Theory 18 (1994), 569–584.

    Article  MathSciNet  MATH  Google Scholar 

  25. Y. Kohayakawa, H. Prömel and V. Rödl: Induced Ramsey numbers, Combinatorica 18 (1998), 373–404.

    Article  MathSciNet  MATH  Google Scholar 

  26. A. V. Kostochka and B. Sudakov: On Ramsey numbers of sparse graphs, Combin. Probab. Comput. 12 (2003), 627–641.

    Article  MathSciNet  MATH  Google Scholar 

  27. M. Krivelevich and B. Sudakov: Pseudorandom graphs, in: More Sets, Graphs and Numbers, Bolyai Soc. Math. Stud. 15, Springer, 2006, 199–262.

  28. L. Lovász: On decomposition of graphs, Studia Sci. Math. Hungar. 1 (1966), 237–238.

    MathSciNet  MATH  Google Scholar 

  29. B. Nagle, S. Olsen, V. Rödl and M. Schacht: On the Ramsey number of sparse 3-graphs, Graphs Combin. 27 (2008), 205–228.

    Article  Google Scholar 

  30. F. P. Ramsey: On a problem of formal logic, Proc. London Math. Soc. Ser. 2 30 (1930), 264–286.

    Article  Google Scholar 

  31. V. Rödl: The dimension of a graph and generalized Ramsey theorems, Master’s thesis, Charles University, 1973.

  32. J. Spencer: Ramsey’s theorem — a new lower bound, J. Combin. Theory Ser. A 18 (1975), 108–115.

    Article  MATH  Google Scholar 

  33. B. Sudakov: A conjecture of Erdős on graph Ramsey numbers, Adv. Math. 227 (2011), 601–609.

    Article  MathSciNet  MATH  Google Scholar 

  34. E. Szemerédi: Regular partitions of graphs, in: Problémes Combinatoires et Théorie des Graphes (Orsay 1976), Colloq. Internat. CNRS, 260, CNRS, Paris, 1978, 399–401.

    Google Scholar 

  35. A. Thomason: Pseudorandom graphs, in: Random graphs’ 85 (Poznań, 1985), North-Holland Math. Stud., Vol. 144, North-Holland, Amsterdam, 1987, 307–331.

    Google Scholar 

  36. A. Thomason: Random graphs, strongly regular graphs and pseudorandom graphs, in: Surveys in Combinatorics 1987, London Math. Soc. Lecture Note Ser., Vol. 123, Cambridge Univ. Press, Cambridge, 1987, 173–195.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. St John’s College, Cambridge, UK

    David Conlon

  2. Department of Mathematics Princeton, Princeton, NJ, USA

    Jacob Fox

  3. Department of Mathematics, UCLA, Los Angeles, CA, 90095, USA

    Benny Sudakov

Authors
  1. David Conlon
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jacob Fox
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Benny Sudakov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to David Conlon.

Additional information

Research supported by a Junior Research Fellowship at St John’s College.

Research supported by a Simons Fellowship, an MIT NEC Corp. award and NSF grant DMS-1069197.

Research supported in part by NSF grant DMS-1101185, by AFOSR MURI grant FA9550-10-1-0569 and by a USA-Israel BSF grant.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Conlon, D., Fox, J. & Sudakov, B. On two problems in graph Ramsey theory. Combinatorica 32, 513–535 (2012). https://doi.org/10.1007/s00493-012-2710-3

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00493-012-2710-3

Mathematics Subject Classification (2000)

Search

Navigation