Upper tails for arithmetic progressions in random subsets



We study the upper tail of the number of arithmetic progressions of a given length in a random subset of {1,..., n}, establishing exponential bounds which are best possible up to constant factors in the exponent. The proof also extends to Schur triples, and, more generally, to the number of edges in random induced subhypergraphs of ‘almost linear’ k-uniform hypergraphs.


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  1. [1]
    N. Alon and J. H. Spencer, The probabilistic method, third ed., Wiley-Interscience Series in Discrete Mathematics and Optimization, John Wiley & Sons, Inc., Hoboken, NJ, 2008.CrossRefGoogle Scholar
  2. [2]
    A. Baltz, P. Hegarty, J. Knape, U. Larsson and T. Schoen. The structure of maximum subsets of {1,...,n} with no solutions to a + b = kc. Electron. J. Combin. 12 (2005), Research Paper 19.Google Scholar
  3. [3]
    S. Boucheron, G. Lugosi and P. Massart. Concentration inequalities using the entropy method. Ann. Probab. 31 (2003), 1583–1614.MathSciNetCrossRefMATHGoogle Scholar
  4. [4]
    S. Boucheron, G. Lugosi and P. Massart, Concentration inequalities, Oxford University Press, Oxford, 2013, A nonasymptotic theory of independence, With a foreword by Michel Ledoux.CrossRefMATHGoogle Scholar
  5. [5]
    S. Chatterjee. The missing log in large deviations for triangle counts. Random Structures Algorithms 40 (2012), 437–451.MathSciNetCrossRefMATHGoogle Scholar
  6. [6]
    S. Chatterjee and P. S. Dey. Applications of Stein’s method for concentration inequalities. Ann. Probab. 38 (2010), 2443–2485.MathSciNetCrossRefMATHGoogle Scholar
  7. [7]
    S. Chatterjee and S. R. S. Varadhan. The large deviation principle for the Erdős-Rényi random graph. European J. Combin. 32 (2011), 1000–1017.MathSciNetCrossRefMATHGoogle Scholar
  8. [8]
    B. Demarco and J. Kahn. Tight upper tail bounds for cliques. Random Structures Algorithms 41 (2012), 469–487.MathSciNetCrossRefMATHGoogle Scholar
  9. [9]
    B. DeMarco and J. Kahn. Upper tails for triangles. Random Structures Algorithms 40 (2012), 452–459.MathSciNetCrossRefMATHGoogle Scholar
  10. [10]
    A. Dembo. Information inequalities and concentration of measure. Ann. Probab. 25 (1997), 927–939.MathSciNetCrossRefMATHGoogle Scholar
  11. [11]
    D. P. Dubhashi and A. Panconesi, Concentration of measure for the analysis of randomized algorithms, Cambridge University Press, Cambridge, 2009.CrossRefMATHGoogle Scholar
  12. [12]
    P. Erdős and P. Tetali. Representations of integers as the sum of k terms. Random Structures Algorithms 1 (1990), 245–261.MathSciNetCrossRefMATHGoogle Scholar
  13. [13]
    E. Friedgut, V. Rödl and M. Schacht. Ramsey properties of random discrete structures. Random Structures Algorithms 37 (2010), 407–436.MathSciNetCrossRefMATHGoogle Scholar
  14. [14]
    R. Graham, V. Rödl and A. Ruciński, On Schur properties of random subsets of integers, J. Number Theory 61 (1996), 388–408.MathSciNetCrossRefMATHGoogle Scholar
  15. [15]
    B. Green. The Cameron-Erdős conjecture. Bull. London Math. Soc. 36 (2004), 769–778.MathSciNetCrossRefMATHGoogle Scholar
  16. [16]
    T. E. Harris, A lower bound for the critical probability in a certain percolation process, Proc. Cambridge Philos. Soc. 56 (1960), 13–20.MathSciNetCrossRefMATHGoogle Scholar
  17. [17]
    S. Janson. Poisson approximation for large deviations. Random Structures Algorithms 1 (1990), 221–229.MathSciNetCrossRefMATHGoogle Scholar
  18. [18]
    S. Janson. New versions of Suen’s correlation inequality, in Proceedings of the Eighth International Conference “Random Structures and Algorithms” (Poznan, 1997). Vol. 13, 1998, pp. 467–483.MathSciNetMATHGoogle Scholar
  19. [19]
    S. Janson, T. Łuczak and A. Rucinski, Random graphs, Wiley-Interscience Series in Discrete Mathematics and Optimization, Wiley-Interscience, New York, 2000.CrossRefGoogle Scholar
  20. [20]
    S. Janson, K. Oleszkiewicz and A. Ruciński, Upper tails for subgraph counts in random graphs, Israel J. Math. 142 (2004), 61–92.MathSciNetCrossRefMATHGoogle Scholar
  21. [21]
    S. Janson and A. Ruciński, The infamous upper tail, Random Structures Algorithms 20 (2002), 317–342, Probabilistic methods in combinatorial optimization.MathSciNetCrossRefMATHGoogle Scholar
  22. [22]
    S. Janson and A. Ruciński, The deletion method for upper tail estimates, Combinatorica 24 (2004), 615–640.MathSciNetCrossRefMATHGoogle Scholar
  23. [23]
    S. Janson and A. Ruciński, Upper tails for counting objects in randomly induced subhypergraphs and rooted random graphs, Ark. Mat. 49 (2011), 79–96.MathSciNetCrossRefMATHGoogle Scholar
  24. [24]
    S. Janson and L. Warnke. The lower tail: Poisson approximation revisited. Random Structures Algorithms 48 (2016), 219–246.MathSciNetCrossRefMATHGoogle Scholar
  25. [25]
    J. H. Kim and V. H. Vu. Concentration of multivariate polynomials and its applications. Combinatorica 20 (2000), 417–434.MathSciNetCrossRefMATHGoogle Scholar
  26. [26]
    M. Ledoux, The concentration of measure phenomenon, Mathematical Surveys and Monographs, Vol. 89, American Mathematical Society, Providence, RI, 2001.Google Scholar
  27. [27]
    E. Lubetzky and Y. Zhao. On replica symmetry of large deviations in random graphs. Random Structures Algorithms 47 (2015), 109–146.MathSciNetCrossRefMATHGoogle Scholar
  28. [28]
    C. McDiarmid, On the method of bounded differences, in Surveys in combinatorics, 1989 (Norwich, 1989), London Math. Soc. Lecture Note Ser., Vol. 141, Cambridge Univ. Press, Cambridge, 1989, pp. 148–188.MathSciNetCrossRefMATHGoogle Scholar
  29. [29]
    C. McDiarmid and B. Reed. Concentration for self-bounding functions and an inequality of Talagrand. Random Structures Algorithms 29 (2006), 549–557.MathSciNetCrossRefMATHGoogle Scholar
  30. [30]
    D. Reimer. Proof of the van den Berg-Kesten conjecture. Combin. Probab. Comput. 9 (2000), 27–32.MathSciNetCrossRefMATHGoogle Scholar
  31. [31]
    O. Riordan and L. Warnke. The Janson inequalities for general up-sets. Random Structures Algorithms 46 (2015), 391–395.MathSciNetCrossRefMATHGoogle Scholar
  32. [32]
    V. Rödl and A. Ruciński, Random graphs with monochromatic triangles in every edge coloring, Random Structures Algorithms 5 (1994), 253–270.MathSciNetCrossRefMATHGoogle Scholar
  33. [33]
    J. Rué and A. Zumalacárregui, Threshold functions for systems of equations on random sets, arXiv:1212.5496 (2012).Google Scholar
  34. [34]
    W. Samotij. Stability results for random discrete structures. Random Structures Algorithms 44 (2014), 269–289.MathSciNetCrossRefMATHGoogle Scholar
  35. [35]
    A. A. Sapozhenko. The Cameron-Erdős conjecture. Dokl. Akad. Nauk 393 (2003), 749–752.MathSciNetGoogle Scholar
  36. [36]
    M. Schacht. Extremal results for random discrete structures. Ann. of Math. (2) 184 (2016), 333–365.MathSciNetCrossRefMATHGoogle Scholar
  37. [37]
    W. Schudy and M. Sviridenko, Concentration and moment inequalities for polynomials of independent random variables, in Proceedings of the Twenty-Third Annual ACM-SIAM Symposium on Discrete Algorithms, ACM, New York, 2012, pp. 437–446.CrossRefGoogle Scholar
  38. [38]
    M. Šileikis, On the upper tail of counts of strictly balanced subgraphs, Electron. J. Combin. 19 (2012), Paper 4, 14.Google Scholar
  39. [39]
    J. Spencer. Counting extensions. J. Combin. Theory Ser. A 55 (1990), 247–255.MathSciNetCrossRefMATHGoogle Scholar
  40. [40]
    M. Talagrand, Concentration of measure and isoperimetric inequalities in product spaces, Inst. Hautes Études Sci. Publ. Math. (1995), 73–205.Google Scholar
  41. [41]
    J. van den Berg and J. Jonasson, A BK inequality for randomly drawn subsets of fixed size, Probab. Theory Related Fields 154 (2012), 835–844.MathSciNetCrossRefMATHGoogle Scholar
  42. [42]
    J. van den Berg and H. Kesten. Inequalities with applications to percolation and reliability. J. Appl. Probab. 22 (1985), 556–569.MathSciNetCrossRefMATHGoogle Scholar
  43. [43]
    V. H. Vu. Concentration of non-Lipschitz functions and applications. Random Structures Algorithms 20 (2002), 262–316, Probabilistic methods in combinatorial optimization.MathSciNetCrossRefMATHGoogle Scholar
  44. [44]
    L. Warnke, On the missing log in upper tail estimates, arXiv:1612.08561 (2016).Google Scholar
  45. [45]
    L. Warnke. When does the K 4-free process stop?. Random Structures Algorithms 44 (2014), 355–397.MathSciNetCrossRefMATHGoogle Scholar
  46. [46]
    L. Warnke. On the method of typical bounded differences. Combin. Probab. Comput. 25 (2016), 269–299.MathSciNetCrossRefGoogle Scholar
  47. [47]
    G. Wolfovitz, A concentration result with application to subgraph count, Random Structures Algorithms 40 (2012), 254–267.MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Hebrew University of Jerusalem 2017

Authors and Affiliations

  1. 1.Department of Pure Mathematics and Mathematical StatisticsUniversity of CambridgeCambridgeUK

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