Inventiones mathematicae

, Volume 181, Issue 2, pp 291–336 | Cite as

The early evolution of the H-free process



The H-free process, for some fixed graph H, is the random graph process defined by starting with an empty graph on n vertices and then adding edges one at a time, chosen uniformly at random subject to the constraint that no H subgraph is formed. Let G be the random maximal H-free graph obtained at the end of the process. When H is strictly 2-balanced, we show that for some c>0, with high probability as n→∞, the minimum degree in G is at least \(cn^{1-(v_{H}-2)/(e_{H}-1)}(\log n)^{1/(e_{H}-1)}\). This gives new lower bounds for the Turán numbers of certain bipartite graphs, such as the complete bipartite graphs K r,r with r≥5. When H is a complete graph K s with s≥5 we show that for some C>0, with high probability the independence number of G is at most \(Cn^{2/(s+1)}(\log n)^{1-1/(e_{H}-1)}\). This gives new lower bounds for Ramsey numbers R(s,t) for fixed s≥5 and t large. We also obtain new bounds for the independence number of G for other graphs H, including the case when H is a cycle. Our proofs use the differential equations method for random graph processes to analyse the evolution of the process, and give further information about the structure of the graphs obtained, including asymptotic formulae for a broad class of subgraph extension variables.


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  1. 1.
    Ajtai, M., Komlós, J., Szemerédi, E.: A note on Ramsey numbers. J. Comb. Theory, Ser. A 29, 354–360 (1980) MATHCrossRefGoogle Scholar
  2. 2.
    Alon, N., Spencer, J.: The Probabilistic Method, second edn. Wiley, New York (2000) MATHGoogle Scholar
  3. 3.
    Alon, N., Rónyai, L., Szabó, T.: Norm-graphs: variations and applications. J. Comb. Theory, Ser. B 76, 280–290 (1999) MATHCrossRefGoogle Scholar
  4. 4.
    Alon, N., Ben-Shimon, S., Krivelevich, M.: A note on regular Ramsey graphs. J. Graph Theory (to appear) Google Scholar
  5. 5.
    Apostol, T.M.: An elementary view of Euler’s summation formula. Am. Math. Mon. 106, 409–418 (1999) MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Barbour, A.D., Karoński, M., Ruciński, A.: A central limit theorem for decomposable random variables, with applications to random graphs. J. Comb. Theory, Ser. B 47, 125–145 (1989) MATHCrossRefGoogle Scholar
  7. 7.
    Bohman, T.: The triangle-free process. Adv. Math. 221, 1653–1677 (2009) MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Bollobás, B., Riordan, O.: Constrained graph processes. Electron. J. Comb. 7, R18 (2000) Google Scholar
  9. 9.
    Brown, W.G.: On graphs that do not contain a Thomsen graph. Can. Math. Bull. 9, 281–289 (1966) MATHGoogle Scholar
  10. 10.
    Caro, Y., Li, Y., Rousseau, C.C., Zhang, Y.: Asymptotic bounds for some bipartite graph—complete graph Ramsey numbers. Discrete Math. 220, 51–56 (2000) MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Durrett, R.: Random Graph Dynamics. Cambridge Univ. Press, Cambridge (2007) MATHGoogle Scholar
  12. 12.
    Erdős, P.: Graph theory and probability, II. Can. J. Math. 13, 346–352 (1961) Google Scholar
  13. 13.
    Erdős, P.: Extremal problems in number theory, combinatorics and geometry. In: Proceedings of the ICM, pp. 51–70. PWN, Warsaw (1984) Google Scholar
  14. 14.
    Erdős, P., Simonovits, M.: Some extremal problems in graph theory. Colloq. Math. Soc. János Bolyai 4, 377–390 (1969) Google Scholar
  15. 15.
    Erdős, P., Spencer, J.H.: Probabilistic Methods in Combinatorics. Academic Press, San Diego (1974) Google Scholar
  16. 16.
    Erdős, P., Stone, A.H.: On the structure of linear graphs. Bull. Am. Math. Soc. 52, 1087–1091 (1946) CrossRefGoogle Scholar
  17. 17.
    Erdős, P., Suen, S., Winkler, P.: On the size of a random maximal graph. Random Struct. Algorithms 6, 309–318 (1995) Google Scholar
  18. 18.
    Füredi, Z.: Turán type problems. In: Surveys in Combinatorics. London Math. Soc. Lecture Note Ser., vol. 166, pp. 253–300. Cambridge Univ. Press, Cambridge (1991) Google Scholar
  19. 19.
    Füredi, Z.: New asymptotics for bipartite Turán numbers. J. Comb. Theory, Ser. A 75, 141–144 (1996) MATHCrossRefGoogle Scholar
  20. 20.
    Füredi, Z.: An upper bound on Zarankiewicz’ problem. Comb. Probab. Comput. 5, 29–33 (1996) MATHCrossRefGoogle Scholar
  21. 21.
    Grable, D.: On random greedy triangle packing. Electron. J. Comb. 4, R11 (1997) MathSciNetGoogle Scholar
  22. 22.
    Graham, R.L., Rothschild, B.L., Spencer, J.H.: Ramsey Theory. Wiley, New York (1990) MATHGoogle Scholar
  23. 23.
    Kim, J.H.: The Ramsey number R(3,t) has order of magnitude t 2/log t. Random Struct. Algorithms 7, 173–207 (1995) MATHCrossRefGoogle Scholar
  24. 24.
    Kövari, T., Sós, V.T., Turán, P.: On a problem of K. Zarankiewicz. Colloq. Math. 3, 50–57 (1954) MATHGoogle Scholar
  25. 25.
    Li, Y., Zang, W.: The independence number of graphs with a forbidden cycle and Ramsey numbers. J. Comb. Optim. 7, 353–359 (2003) MATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    Osthus, D., Taraz, A.: Random maximal H-free graphs. Random Struct. Algorithms 18, 61–82 (2001) MATHCrossRefMathSciNetGoogle Scholar
  27. 27.
    Ruciński, A.: Recent developments in random graphs. In: Proceedings of the International Summer School on Probability and Statistics, Varna. Online at: (1994)
  28. 28.
    Ruciński, A., Wormald, N.: Random graph processes with degree restrictions. Comb. Probab. Comput. 1, 169–180 (1992) MATHGoogle Scholar
  29. 29.
    Seierstad, T.G.: A central limit theorem via differential equations. Ann. Appl. Probab. 19, 661–675 (2009) MATHCrossRefMathSciNetGoogle Scholar
  30. 30.
    Seierstad, T.G.: Stronger large deviation bounds for Wormald’s differential equation method (submitted) Google Scholar
  31. 31.
    Spencer, J.: Counting extensions. J. Comb. Theory, Ser. A 55, 247–255 (1990) MATHCrossRefGoogle Scholar
  32. 32.
    Spencer, J.: Asymptotic lower bounds for Ramsey functions. Discrete Math. 20, 69–76 (1997) CrossRefGoogle Scholar
  33. 33.
    Spencer, J.: Maximal trianglefree graphs and Ramsey R(3,k). Available online at: (unpublished manuscript)
  34. 34.
    Sudakov, B.: A note on odd cycle-complete graph Ramsey numbers. Electron. J. Comb. 9, N1 (2002) MathSciNetGoogle Scholar
  35. 35.
    Turán, P.: Eine Extremalaufgabe aus der Graphentheorie. Mat. Fiz. Lapok 48, 436–452 (1941) MATHMathSciNetGoogle Scholar
  36. 36.
    Wolfovitz, G.: Lower bounds for the size of random maximal H-free graphs. Electron. J. Comb. 16, R4 (2009) MathSciNetGoogle Scholar
  37. 37.
    Wormald, N.C.: The differential equation method for random graph processes and greedy algorithms. In: Lectures on Approximation and Randomized Algorithms, pp. 73–155. PWN, Warsaw (1999) Google Scholar

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© Springer-Verlag 2010

Authors and Affiliations

  1. 1.Department of Mathematical SciencesCarnegie Mellon UniversityPittsburghUSA
  2. 2.School of Mathematical SciencesQueen Mary, University of LondonLondonUK

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