The Random Walk Method for Intersecting Families

  • Norihide Tokushige
Part of the Bolyai Society Mathematical Studies book series (BSMS, volume 17)


Let m(n, k, r, t) be the maximum size of Open image in new window satisfying |F 1 ∩ ... ∩ F r| ≥t for all F 1,...,F r ∈ ℱ. We report some known results about m(n,k,r,t). The random walk method introduced by Frankl is a strong tool to investigate m(n, k, r, t). Using a concrete example, we explain the method and how to use it.


Intersection Theorem Random Walk Method BOLYAI Society Mathematical BOLYAI Society Intersecting Family 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    R. Ahlswede and L. H. Khachatrian, The complete nontrivial-intersection theorem for systems of finite sets, J. Combin. Theory (A), 76 (1996), 121–138.zbMATHCrossRefMathSciNetGoogle Scholar
  2. [2]
    R. Ahlswede and L. H. Khachatrian, The complete intersection theorem for systems of finite sets, European J. Combin., 18 (1997), 125–136.zbMATHCrossRefMathSciNetGoogle Scholar
  3. [3]
    R. Ahlswede and L. H. Khachatrian, The diametric theorem in Hamming spaces — Optimal anticodes, Adv. in Appl. Math., 20 (1998), 429–449.zbMATHCrossRefMathSciNetGoogle Scholar
  4. [4]
    A. Brace and D. E. Daykin, A finite set covering theorem, Bull. Austral. Math. Soc., 5 (1971), 197–202.zbMATHMathSciNetCrossRefGoogle Scholar
  5. [5]
    C. Bey and K. Engel, Old and new results for the weighted t-intersection problem via AK-methods, Numbers, Information and Complexity, Althofer, Ingo, Eds. et al., Dordrecht, Kluwer Academic Publishers (2000), pp. 45–74.Google Scholar
  6. [6]
    P. Erdős, C. Ko and R. Rado, Intersection theorems for systems of finite sets, Quart. J. Math. Oxford (2), 12 (1961), 313–320.CrossRefGoogle Scholar
  7. [7]
    I. Dinur and S. Safra, On the Hardness of Approximating Minimum Vertex-Cover, Annals of Mathematics, 162 (2005), 439–485.zbMATHMathSciNetGoogle Scholar
  8. [8]
    P. Frankl and On Sperner families satisfying an additional condition, J. Combin. Theory (A), 20 (1976), 1–11.zbMATHCrossRefMathSciNetGoogle Scholar
  9. [9]
    . P. Frankl, Families of finite sets satisfying an intersection condition, Bull. Austral. Math. Soc., 15 (1976), 73–79.zbMATHMathSciNetGoogle Scholar
  10. [10]
    P. Prankl, The Erdős-Ko-Rado theorem is true for n = ckt, Combinatorics (Proc. Fifth Hungarian Colloq., Keszthely, 1976), Vol. I, 365–375, Colloq. Math. Soc. János Bolyai, 18, North-Holland (1978).Google Scholar
  11. [11]
    P. Frankl, On intersecting families of finite sets, J. Combin. Theory (A), 24 (1978), 141–161.CrossRefMathSciNetGoogle Scholar
  12. [12]
    P. Frankl, The shifting technique in extremal set theory, “Surveys in Combinatorics 1987” (C. Whitehead, Ed. LMS Lecture Note Series 123), 81–110, Cambridge Univ. Press (1987).Google Scholar
  13. [13]
    P. Frankl, Multiply-intersecting families, J. Combin. Theory (B), 53 (1991), 195–234.zbMATHCrossRefMathSciNetGoogle Scholar
  14. [14]
    P. Frankl and N. Tokushige, Weighted 3-wise 2-intersecting families, J. Combin. Theory (A), 100 (2002), 94–115.zbMATHCrossRefMathSciNetGoogle Scholar
  15. [15]
    P. Frankl and N. Tokushige, Weighted multiply intersecting families, Studia Sci. Math. Hungarica, 40 (2003), 287–291.zbMATHCrossRefMathSciNetGoogle Scholar
  16. [16]
    P. Frankl and N. Tokushige, Random walks and multiply intersecting families, J. Combin. Theory (A), 109 (2005), 121–134.zbMATHCrossRefMathSciNetGoogle Scholar
  17. [17]
    P. Frankl and N. Tokushige, Weighted non-trivial multiply intersecting families, Combinatorica, 26 (2006), 37–46.zbMATHCrossRefMathSciNetGoogle Scholar
  18. [18]
    A. J. W. Hilton and E. C. Milner, Some intersection theorems for systems of finite sets, Quart. J. Math. Oxford, 18 (1967), 369–384.zbMATHCrossRefMathSciNetGoogle Scholar
  19. [19]
    G. O. H. Katona, Intersection theorems for systems of finite sets, Acta Math. Acad. Sci. Hung., 15 (1964), 329–337.zbMATHCrossRefMathSciNetGoogle Scholar
  20. [20]
    G. O. H. Katona, A theorem of finite sets, in: Theory of Graphs, Proc. Colloq. Tihany, 1966 (Akadémiai Kiadó, 1968), pp. 187–207.Google Scholar
  21. [21]
    J. B. Kruskal, The number of simplices in a complex, in: Math. Opt Techniques (Univ. of Calif. Press, 1963), pp. 251–278.Google Scholar
  22. [22]
    N. Tokushige, A frog’s random jump and the Pólya identity, Ryukyu Math. Journal, 17 (2004), 89–103.zbMATHMathSciNetGoogle Scholar
  23. [23]
    N. Tokushige, Intersecting families — uniform versus weighted, Ryukyu Math. J., 18 (2005), 89–103.zbMATHMathSciNetGoogle Scholar
  24. [24]
    N. Tokushige, Extending the Erdős-Ko-Rado theorem, J. Combin. Designs, 14 (2006), 52–55.CrossRefMathSciNetGoogle Scholar
  25. [25]
    N. Tokushige, The maximum size of 4-wise 2-intersecting and 4-wise 2-union families, European J. of Comb., 27 (2006), 814–825.zbMATHCrossRefMathSciNetGoogle Scholar
  26. [26]
    N. Tokushige, The maximum size of 3-wise t-intersecting families, European J. Combin, 28 (2007), 152–166.zbMATHCrossRefMathSciNetGoogle Scholar
  27. [27]
    N. Tokushige, EKR type inequalities for 4-wise intersecting families, J. Combin. Theory (A), 114 (2007), 575–596.zbMATHCrossRefMathSciNetGoogle Scholar
  28. [28]
    N. Tokushige, Brace-Daykin type inequalities for intersecting families, European J. Combin, 29 (2008), 273–285.zbMATHCrossRefMathSciNetGoogle Scholar
  29. [29]
    N. Tokushige, Multiply-intersecting families revisited, J. Combin. Theory (B), 97 (2007), 929–948.zbMATHCrossRefMathSciNetGoogle Scholar
  30. [30]
    R. M. Wilson, The exact bound in the Erdős-Ko-Rado theorem, Combinatorica, 4 (1984), 247–257.zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© János Bolyai Mathematical Society and Springer-Verlag 2008

Authors and Affiliations

  • Norihide Tokushige
    • 1
  1. 1.College of EducationRyukyu UniversityNishihara, OkinawaJapan

Personalised recommendations