Packing Nearly Optimal Ramsey R(3,t) Graphs


In 1995 Kim famously proved the Ramsey bound R(3, t) ≤ ct2/logt by constructing an n-vertex graph that is triangle-free and has independence number at most \(C\,\sqrt {n\,\log \,n} \). We extend this celebrated result, which is best possible up to the value of the constants, by approximately decomposing the complete graph Kn into a packing of such nearly optimal Ramsey R(3,t) graphs.

More precisely, for any ε > 0 we find an edge-disjoint collection (Gi)i of n-vertex graphs GiKn such that (a) each Gi is triangle-free and has independence number at most \(C_{\epsilon} \sqrt{n \log n}\), and (b) the union of all the Gi contains at least \((1-\epsilon)\left(\begin{array}{l}n \\2\end{array}\right)\) edges. Our algorithmic proof proceeds by sequentially choosing the graphs Gi via a semi-random (i.e., Rödl nibble type) variation of the triangle-free process.

As an application, we prove a conjecture in Ramsey theory by Fox, Grinshpun, Liebenau, Person, and Szabó (concerning a Ramsey-type parameter introduced by Burr, Erdős, and Lovász in 1976). Namely, denoting by sr(H) the smallest minimum degree of r-Ramsey minimal graphs for H, we close the existing logarithmic gap for H = K3 and establish that sr(K3) = Θ(r2 logr).

This is a preview of subscription content, access via your institution.


  1. [1]

    M. Ajtai, J. Komlós and E. Szemerédi: A note on Ramsey numbers, J. Combin. Theory Ser. A29 (1980), 354–360.

    MathSciNet  Article  Google Scholar 

  2. [2]

    M. Ajtai, J. Komlós and E. Szemerédi: A dense infinite Sidon sequence, European J. Combin.2 (1981), 1–11.

    MathSciNet  Article  Google Scholar 

  3. [3]

    N. Alon, J. H. Kim and J. Spencer: Nearly perfect matchings in regular simple hypergraphs, Israel J. Math.100 (1997), 171–187.

    MathSciNet  Article  Google Scholar 

  4. [4]

    J. Beck: On size Ramsey number of paths, trees, and circuits. I, J. Graph Theory7 (1983), 115–129.

    MathSciNet  Article  Google Scholar 

  5. [5]

    T. Bohman: The triangle-free process, Adv. Math.221 (2009), 1653–1677.

    MathSciNet  Article  Google Scholar 

  6. [6]

    T. Bohman and P. Keevash: The early evolution of the H-free process, Invent. Math.181 (2010), 291–336.

    MathSciNet  Article  Google Scholar 

  7. [7]

    T. Bohman and P. Keevash: Dynamic concentration of the triangle-free process, Preprint (2013). arXiv:1302.5963.

  8. [8]

    S. A. Burr, P. Erdős and L. Lovász: On graphs of Ramsey type, Ars Combinatoria1 (1976), 167–190.

    MathSciNet  MATH  Google Scholar 

  9. [9]

    D. Conlon, J. Fox and B. Sudakov: Recent developments in graph Ramsey theory, in: Surveys in Combinatorics 2015, 49–118. Cambridge Univ. Press, Cambridge (2015).

    Google Scholar 

  10. [10]

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

    MathSciNet  Article  Google Scholar 

  11. [11]

    P. Erdős: Graph theory and probability. II, Canad. J. Math.13 (1961), 346–352.

    MathSciNet  Article  Google Scholar 

  12. [12]

    P. Erdős, R. J. Faudree, C. C. Rousseau and R. H. Schelp: The size Ramsey number, Period. Math. Hungar.9 (1978), 145–161.

    MathSciNet  Article  Google Scholar 

  13. [13]

    P. Erdős, S. Suen and P. Winkler: On the size of a random maximal graph, Rand. Struct. & Algor.6 (1995), 309–318.

    MathSciNet  Article  Google Scholar 

  14. [14]

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

    MathSciNet  MATH  Google Scholar 

  15. [15]

    G. Fiz Pontiveros, S. Griffiths and R. Morris: The triangle-free process and R(3, k), Memoirs of the Am. Math. Soc., to appear.

  16. [16]

    J. Fox, A. Grinshpun, A. Liebenau, Y. Person and T. Szabó: On the minimum degree of minimal Ramsey graphs for multiple colours, J. Combin. Theory Ser. B120 (2016), 64–82.

    MathSciNet  Article  Google Scholar 

  17. [17]

    J. Fox and K. Lin: The minimum degree of Ramsey-minimal graphs, J. Graph Theory54 (2007), 167–177.

    MathSciNet  Article  Google Scholar 

  18. [18]

    T. E. Harris: A lower bound for the critical probability in a certain percolation process, Proc. Cambridge Philos. Soc.56 (1960), 13–20.

    MathSciNet  Article  Google Scholar 

  19. [19]

    J. Kahn: Asymptotically good list-colorings, J. Combin. Theory Ser. A73 (1996), 1–59.

    MathSciNet  Article  Google Scholar 

  20. [20]

    J. H. Kim: The Ramsey number R(3, t) has order of magnitude t2/logt, Rand. Struct. & Algor.7 (1995), 173–207.

    Article  Google Scholar 

  21. [21]

    M. Krivelevich: Bounding Ramsey numbers through large deviation inequalities, Rand. Struct. & Algor.7 (1995), 145–155.

    MathSciNet  Article  Google Scholar 

  22. [22]

    A. Liebenau: Orientation Games and Minimal Ramsey Graphs, PhD thesis, FU Berlin (2013).

  23. [23]

    C. McDiarmid: On the method of bounded differences, in: Surveys in Combinatorics 1989, 148–188. Cambridge Univ. Press, Cambridge (1989).

    Google Scholar 

  24. [24]

    C. McDiarmid: Concentration, in: Probabilistic methods for algorithmic discrete mathematics, 195–248. Springer, Berlin (1998).

    Google Scholar 

  25. [25]

    D. Osthus and A. Taraz: Random maximal H-free graphs, Rand. Struct. & Algor.18 (2001), 61–82.

    MathSciNet  Article  Google Scholar 

  26. [26]

    Y. Person: Personal communication (RSA 2013 conference in Poznań), 2013.

  27. [27]

    M. Picollelli: The diamond-free process, Rand. Struct. & Algor.45 (2014), 513–551.

    MathSciNet  Article  Google Scholar 

  28. [28]

    F. P. Ramsey: On a Problem of Formal Logic, Proc. Lond. Math. Soc.30 (1930), 264–286.

    MathSciNet  Article  Google Scholar 

  29. [29]

    V. Rödl and M. Siggers: On Ramsey minimal graphs, SIAM J. Discrete Math.22 (2008), 467–488.

    MathSciNet  Article  Google Scholar 

  30. [30]

    V. Rödl and E. Szemerédi: On size Ramsey numbers of graphs with bounded degree, Combinatorica20 (2000), 257–262.

    MathSciNet  Article  Google Scholar 

  31. [31]

    J. Spencer: Asymptotic lower bounds for Ramsey functions, Discrete Math.20 (1977), 69–76.

    MathSciNet  Article  Google Scholar 

  32. [32]

    J. Spencer: Maximal triangle-free graphs and Ramsey R(3, t), Unpublished manuscript (1995).

  33. [33]

    T. Szabó, P. Zumstein and S. Zürcher: On the minimum degree of minimal Ramsey graphs, J. Graph Theory64 (2010), 150–164.

    MathSciNet  MATH  Google Scholar 

  34. [34]

    L. Warnke: When does the K4-free process stop? Rand. Struct. & Algor.44 (2014), 355–397.

    MathSciNet  Article  Google Scholar 

  35. [35]

    L. Warnke: The C-free process, Rand. Struct. & Algor.44 (2014), 490–526.

    MathSciNet  Article  Google Scholar 

  36. [36]

    L. Warnke: On the method of typical bounded differences, Combin. Probab. Comput.25 (2016), 269–299.

    MathSciNet  Article  Google Scholar 

  37. [37]

    L. Warnke: Upper tails for arithmetic progressions in random subsets, Israel J. Math.221 (2017), 317–365.

    MathSciNet  Article  Google Scholar 

  38. [38]

    G. Wolfovitz: Triangle-free subgraphs in the triangle-free process, Rand. Struct. & Algor.39 (2011), 539–543.

    MathSciNet  Article  Google Scholar 

Download references

Author information



Corresponding authors

Correspondence to He Guo or Lutz Warnke.

Additional information

Research partially supported by NSF Grant DMS-1703516 and a Sloan Research Fellowship.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Guo, H., Warnke, L. Packing Nearly Optimal Ramsey R(3,t) Graphs. Combinatorica 40, 63–103 (2020).

Download citation

Mathematics Subject Classification (2010)

  • 05C55
  • 05C80
  • 05D10
  • 60C05