The critical window for the classical Ramsey-Turán problem

Abstract

The first application of Szemerédi’s powerful regularity method was the following celebrated Ramsey-Turán result proved by Szemerédi in 1972: any K 4-free graph on n vertices with independence number o(n) has at most \((\tfrac{1} {8} + o(1))n^2\) edges. Four years later, Bollobás and Erdős gave a surprising geometric construction, utilizing the isoperimetric inequality for the high dimensional sphere, of a K 4-free graph on n vertices with independence number o(n) and \((\tfrac{1} {8} - o(1))n^2\) edges. Starting with Bollobás and Erdős in 1976, several problems have been asked on estimating the minimum possible independence number in the critical window, when the number of edges is about n 2/8. These problems have received considerable attention and remained one of the main open problems in this area. In this paper, we give nearly best-possible bounds, solving the various open problems concerning this critical window.

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References

  1. [1]

    N. Alon and J. Spencer: The probabilistic method, 3rd ed., John Wiley & Sons Inc., Hoboken, NJ (2008).

    Google Scholar 

  2. [2]

    K. Ball: An elementary introduction to modern convex geometry, in: Flavors of geometry, Math. Sci. Res. Inst. Publ. 31, Cambridge Univ. Press, Cambridge, 1997, 1–58.

    Google Scholar 

  3. [3]

    J. Balogh and J. Lenz: On the Ramsey-Turán numbers of graphs and hypergraphs, Israel J. Math. 194 (2013), 45–68.

    Article  MathSciNet  MATH  Google Scholar 

  4. [4]

    J. Balogh and J. Lenz: Some exact Ramsey-Turán numbers, Bull. Lond. Math. Soc. 44 (2012), 1251–1258.

    Article  MathSciNet  MATH  Google Scholar 

  5. [5]

    B. Bollobás and P. Erdős: On a Ramsey-Turán type problem, J. Combin. Theory Ser. B. 21 (1976), 166–168.

    Article  MATH  Google Scholar 

  6. [6]

    C. Borgs, J. T. Chayes, L. Lovász, V. T. Sós and K. Vesztergombi: Convergent sequences of dense graphs. I. Subgraph frequencies, metric properties and testing, Adv. Math. 219 (2008), 1801–1851.

    Article  MathSciNet  MATH  Google Scholar 

  7. [7]

    P. Châu, L. DeBiasio and H. A. Kierstead: Pósa’s conjecture for graphs of order at least 2 × 108, Random Structures Algorithms 39 (2011), 507–525.

    Article  MathSciNet  MATH  Google Scholar 

  8. [8]

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

    Article  MathSciNet  MATH  Google Scholar 

  9. [9]

    D. Conlon and J. Fox: Bounds for graph regularity and removal lemmas, Geom. Funct. Anal. 22 (2012), 1191–1256.

    Article  MathSciNet  MATH  Google Scholar 

  10. [10]

    D. Conlon, J. Fox and B. Sudakov: On two problems in graph Ramsey theory, Combinatorica 32 (2012), 513–535.

    Article  MathSciNet  MATH  Google Scholar 

  11. [11]

    P. Erdős: Some recent results on extremal problems in graph theory. Results, in: Theory of Graphs (Internat. Sympos., Rome, 1966), Gordon and Breach, New York, 1967, 117–123.

    Google Scholar 

  12. [12]

    P. Erdős: Some of my favourite unsolved problems, in: A tribute to Paul Erdős, Cambridge University Press, 1990, 467–478.

    Google Scholar 

  13. [13]

    P. Erdős, A. Hajnal, V. T. Sós and E. Szemerédi: More results on Ramsey-Turán type problem, Combinatorica 3 (1983), 69–82.

    Article  MathSciNet  Google Scholar 

  14. [14]

    P. Erdős, A. Hajnal, M. Simonovits, V. T. Sós and E. Szemerédi: Turán-Ramsey theorems and simple asymptotically extremal structures, Combinatorica 13 (1993), 31–56.

    Article  MathSciNet  Google Scholar 

  15. [15]

    P. Erdős, A. Hajnal, M. Simonovits, V. T. Sós and E. Szemerédi: Turán- Ramsey theorems and K p-independence number, in: Combinatorics, geometry and probability (Cambridge, 1993), Cambridge Univ. Press, Cambridge, 1997, 253–281.

    Google Scholar 

  16. [16]

    P. Erdős and V. T. Sós: Some remarks on Ramsey’s and Turán’s theorem, in: Combinatorial theory and its applications, II (Proc. Colloq., Balatonfüred, 1969), North-Holland, Amsterdam, 1970, 395–404.

    Google Scholar 

  17. [17]

    U. Feige and G. Schechtman: On the optimality of the random hyperplane rounding technique for MAX CUT, Random Structures Algorithms 20 (2002), 403–440.

    Article  MathSciNet  MATH  Google Scholar 

  18. [18]

    J. Fox: A new proof of the graph removal lemma, Ann. of Math. 174 (2011), 561–579.

    Article  MathSciNet  MATH  Google Scholar 

  19. [19]

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

    MathSciNet  MATH  Google Scholar 

  20. [20]

    J. Fox and B. Sudakov: Dependent random choice, Random Structures Algorithms 38 (2011), 68–99.

    Article  MathSciNet  MATH  Google Scholar 

  21. [21]

    A. Frieze and R. Kannan: The regularity lemma and approximation schemes for dense problems, Proceedings of the 37th IEEE FOCS (1996), 12–20.

    Google Scholar 

  22. [22]

    A. Frieze and R. Kannan: Quick approximation to matrices and applications, Combinatorica 19 (1999), 175–220.

    Article  MathSciNet  MATH  Google Scholar 

  23. [23]

    A. W. Goodman: On sets of acquaintances and strangers at any party, American Mathematical Monthly 66 (1959), 778–783.

    Article  MathSciNet  MATH  Google Scholar 

  24. [24]

    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 

  25. [25]

    W. T. Gowers: Rough structure and classification, GAFA 2000 (Tel Aviv, 1999), Geom. Funct. Anal. 2000, Special Volume, Part I, 79–117.

    Google Scholar 

  26. [26]

    W. T. Gowers: A new proof of Szemerédi’s theorem, Geom. Funct. Anal. 11 (2001), 465–588.

    Article  MathSciNet  MATH  Google Scholar 

  27. [27]

    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 

  28. [28]

    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, 295–352.

    Google Scholar 

  29. [29]

    J. Komlós, A. Shokoufandeh, M. Simonovits and E. Szemerédi: The regularity lemma and its applications in graph theory, in: Theoretical aspects of computer science (Tehran, 2000), Lecture Notes in Comput. Sci., 2292, Springer, Berlin, 2002, 84–112.

    Google Scholar 

  30. [30]

    I. Levitt, G. N. Sárközy and E. Szemerédi: How to avoid using the regularity lemma: Pósa’s conjecture revisited, Discrete Math. 310 (2010), 630–641.

    Article  MathSciNet  MATH  Google Scholar 

  31. [31]

    V. Rödl and M. Schacht: Regularity lemmas for graphs, in: Fete of Combinatorics and Computer Science, Bolyai Soc. Math. Stud. 20, 2010, 287–325.

    Google Scholar 

  32. [32]

    I. Z. Ruzsa and E. Szemerédi: Triple systems with no six points carrying three triangles, in: Combinatorics (Keszthely, 1976), Coll. Math. Soc. J. Bolyai 18, Volume II, 939–945.

    Google Scholar 

  33. [33]

    E. Schmidt: Die Brunn-Minkowski Ungleichung und ihr Spiegelbild sowie die isoperimetrische Eigenschaft der Kugel in der euklidischen und nichteuklidischen Geometrie I, Math Nachrichten 1 (1948), 81–157.

    Article  MATH  Google Scholar 

  34. [34]

    J. B. Shearer: The independence number of dense graphs with large odd girth, Electron. J. Combin. 2 (1995), Note 2 (electronic).

  35. [35]

    M. Simonovits: A method for solving extremal problems in graph theory, stability problems, in: Theory of Graphs (Proc. Colloq., Tihany, 1966), Academic Press, New York, 1968, 279–319.

    Google Scholar 

  36. [36]

    M. Simonovits and V. T. Sós: Ramsey-Turán theory, Discrete Math. 229 (2001), 293–340.

    Article  MathSciNet  MATH  Google Scholar 

  37. [37]

    V. T. Sós: On extremal problems in graph theory, in: Proceedings of the Calgary International Conference on Combinatorial Structures and their Application, 1969, 407–410.

    Google Scholar 

  38. [38]

    B. Sudakov: A few remarks on Ramsey-Turán-type problems, J. Combin. Theory Ser. B 88 (2003), 99–106.

    Article  MathSciNet  MATH  Google Scholar 

  39. [39]

    E. Szemerédi: On graphs containing no complete subgraph with 4 vertices (Hungarian), Mat. Lapok 23 (1972), 113–116.

    MathSciNet  Google Scholar 

  40. [40]

    E. Szemerédi: On sets of integers containing no κ elements in arithmetic progression, Acta Arith. 27 (1975), 199–245.

    MathSciNet  MATH  Google Scholar 

  41. [41]

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

    Google Scholar 

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Correspondence to Po-Shen Loh.

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Research supported by a Simons Fellowship, NSF grant DMS-1069197, and the MIT NEC Research Corporation Fund.

Research supported by NSF grant DMS-1201380, an NSA Young Investigators Grant and a USA-Israel BSF Grant.

Research supported by an Akamai Presidential Fellowship.

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Fox, J., Loh, PS. & Zhao, Y. The critical window for the classical Ramsey-Turán problem. Combinatorica 35, 435–476 (2015). https://doi.org/10.1007/s00493-014-3025-3

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  • 05C55
  • 05D40