Israel Journal of Mathematics

, Volume 221, Issue 1, pp 317–365 | Cite as

Upper tails for arithmetic progressions in random subsets

  • Lutz WarnkeEmail author


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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  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.Google 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.MathSciNetCrossRefzbMATHGoogle 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 byMichel Ledoux.CrossRefzbMATHGoogle Scholar
  5. [5]
    S. Chatterjee, The missing log in large deviations for triangle counts, Random Structures Algorithms 40 (2012), 437–451.MathSciNetCrossRefzbMATHGoogle Scholar
  6. [6]
    S. Chatterjee and P. S. Dey, Applications of Stein’s method for concentration inequalities, Ann. Probab. 38 (2010), 2443–2485.MathSciNetCrossRefzbMATHGoogle 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.MathSciNetCrossRefzbMATHGoogle Scholar
  8. [8]
    B. Demarco and J. Kahn, Tight upper tail bounds for cliques, Random Structures Algorithms 41 (2012), 469–487.MathSciNetCrossRefzbMATHGoogle Scholar
  9. [9]
    B. DeMarco and J. Kahn, Upper tails for triangles, Random Structures Algorithms 40 (2012), 452–459.MathSciNetCrossRefzbMATHGoogle Scholar
  10. [10]
    A. Dembo, Information inequalities and concentration of measure, Ann. Probab. 25 (1997), 927–939.MathSciNetCrossRefzbMATHGoogle Scholar
  11. [11]
    D. P. Dubhashi and A. Panconesi, Concentration of measure for the analysis of randomized algorithms, Cambridge University Press, Cambridge, 2009.CrossRefzbMATHGoogle Scholar
  12. [12]
    P. Erdős and P. Tetali, Representations of integers as the sum of k terms, Random Structures Algorithms 1 (1990), 245–261.MathSciNetCrossRefzbMATHGoogle Scholar
  13. [13]
    E. Friedgut, V. Rödl and M. Schacht, Ramsey properties of random discrete structures, Random Structures Algorithms 37 (2010), 407–436.MathSciNetCrossRefzbMATHGoogle 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.MathSciNetCrossRefzbMATHGoogle Scholar
  15. [15]
    B. Green, The Cameron-Erdős conjecture, Bull. London Math. Soc. 36 (2004), 769–778.MathSciNetCrossRefzbMATHGoogle 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.MathSciNetCrossRefzbMATHGoogle Scholar
  17. [17]
    S. Janson, Poisson approximation for large deviations, Random Structures Algorithms 1 (1990), 221–229.MathSciNetCrossRefzbMATHGoogle 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.Google 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.Google 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.MathSciNetCrossRefzbMATHGoogle Scholar
  21. [21]
    S. Janson and A. Ruciński, The infamous upper tail, Random Structures Algorithms 20 (2002), 317–342, Probabilistic methods in combinatorial optimization.MathSciNetCrossRefzbMATHGoogle Scholar
  22. [22]
    S. Janson and A. Ruciński, The deletion method for upper tail estimates, Combinatorica 24 (2004), 615–640.MathSciNetCrossRefzbMATHGoogle 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.MathSciNetCrossRefzbMATHGoogle Scholar
  24. [24]
    S. Janson and L. Warnke, The lower tail: Poisson approximation revisited, Random Structures Algorithms 48 (2016), 219–246.MathSciNetCrossRefzbMATHGoogle Scholar
  25. [25]
    J. H. Kim and V. H. Vu, Concentration of multivariate polynomials and its applications, Combinatorica 20 (2000), 417–434.MathSciNetCrossRefzbMATHGoogle Scholar
  26. [26]
    M. Ledoux, The concentration of measure phenomenon, Mathematical Surveys and Monographs, Vol. 89, American Mathematical Society, Providence, RI, 2001.zbMATHGoogle Scholar
  27. [27]
    E. Lubetzky and Y. Zhao, On replica symmetry of large deviations in random graphs, Random Structures Algorithms 47 (2015), 109–146.MathSciNetCrossRefzbMATHGoogle 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.CrossRefGoogle 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.MathSciNetCrossRefzbMATHGoogle Scholar
  30. [30]
    D. Reimer, Proof of the van den Berg-Kesten conjecture, Combin. Probab. Comput. 9 (2000), 27–32.MathSciNetCrossRefzbMATHGoogle Scholar
  31. [31]
    O. Riordan and L. Warnke, The Janson inequalities for general up-sets, Random Structures Algorithms 46 (2015), 391–395.MathSciNetCrossRefzbMATHGoogle Scholar
  32. [32]
    V. Rödl and A. Ruciński, Random graphs with monochromatic triangles in every edge coloring, Random Structures Algorithms 5 (1994), 253–270.MathSciNetCrossRefzbMATHGoogle 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.MathSciNetCrossRefzbMATHGoogle 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.MathSciNetCrossRefzbMATHGoogle 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.MathSciNetzbMATHGoogle Scholar
  39. [39]
    J. Spencer, Counting extensions, J. Combin. Theory Ser. A 55 (1990), 247–255.MathSciNetCrossRefzbMATHGoogle 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.MathSciNetCrossRefzbMATHGoogle Scholar
  42. [42]
    J. van den Berg and H. Kesten, Inequalities with applications to percolation and reliability, J. Appl. Probab. 22 (1985), 556–569.MathSciNetCrossRefzbMATHGoogle 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.MathSciNetCrossRefzbMATHGoogle 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.MathSciNetCrossRefzbMATHGoogle 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.MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Hebrew University of Jerusalem 2017

Authors and Affiliations

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

Personalised recommendations