On sperner families in which nok sets have an empty intersection III

Abstract

LetR be anr-element set and ℱ be a Sperner family of its subsets, that is,XY for all differentX, Y ∈ ℱ. The maximum cardinality of ℱ is determined under the conditions 1)c≦|X|≦d for allX ∈ ℱ, (c andd are fixed integers) and 2) nok sets (k≧4, fixed integer) in ℱ have an empty intersection. The result is mainly based on a theorem which is proved by induction, simultaneously with a theorem of Frankl.

This is a preview of subscription content, access via your institution.

References

  1. [1]

    G. F. Clements, A minimization problem concerning subsets,Discrete Math.4 (1973), 123–128.

    MATH  Article  MathSciNet  Google Scholar 

  2. [2]

    D. E. Daykin, Jean Godfrey andA. J. W. Hilton, Existence theorems for Sperner families,J. Combin. Theory (A)17 (1974), 245–251.

    MATH  Article  MathSciNet  Google Scholar 

  3. [3]

    M. Deza, P. Erdős andP. Frankl, Intersection properties of systems of finite sets,Proc. London Math. Soc. (3)36 (1978), 369–384.

    MATH  Article  MathSciNet  Google Scholar 

  4. [4]

    P. Erdős, Chao Ko andP. Rado, Intersection theorems for systems of finite sets,Quart. J. Math. Oxford (2)12 (1961), 313–320.

    Article  Google Scholar 

  5. [5]

    P. Frankl, On Sperner families satisfying an additional condition,J. Combin. Theory (A)20 (1976), 1–11.

    MATH  Article  MathSciNet  Google Scholar 

  6. [6]

    P. Frankl, Extremal problems and coverings of the space,European J. of Combinatorics1 (1980), 101–106.

    MATH  MathSciNet  Google Scholar 

  7. [7]

    P. Frankl, Families of finite sets with prescribed cardinalities for pairwise intersections,Acta Math. Acad. Hungar.35 (1980), 351–360.

    MATH  Article  MathSciNet  Google Scholar 

  8. [8]

    C. Greene andA. J. W. Hilton, Some results on Sperner families,J. Combin. Theory (A),26 (1979), 202–209.

    MATH  Article  MathSciNet  Google Scholar 

  9. [9]

    H.-D. O. F. Gronau, On Sperner families in which nok sets have an empty intersection,J. Combin. Theory (A),28 (1980), 54–63.

    MATH  Article  MathSciNet  Google Scholar 

  10. [10]

    H.-D. O. F. Gronau, On Sperner families in which no 3 sets have an empty intersection,Acta Cybernetica,4 (1978), 213–220.

    MathSciNet  Google Scholar 

  11. [11]

    H.-D. O. F. Gronau, On Sperner families in which nok sets have an empty intersection II,J. Combin. Theory (A)30 (1981), 298–316.

    MATH  Article  MathSciNet  Google Scholar 

  12. [12]

    H.-D. O. F. Gronau, Sperner type theorems and complexity of minimal disjunctive normal forms of monotone Boolean function,Period. Math. Hung.,12 (4) (1981) 267–282.

    MATH  Article  MathSciNet  Google Scholar 

  13. [13]

    A. J. W. Hilton andE. C. Milner, Some intersection theorems for system of finite sets,Quart. J. Math. Oxford 2,18 (1967), 369–384.

    MATH  Article  MathSciNet  Google Scholar 

  14. [14]

    G. Katona, A theorem for finite sets,Theory of Graphs, Proc. Colloq., Tihany. Hungary (1966), 187–207.

  15. [15]

    J. B. Kruskal, The number of simplices in a complex, in:Mathematical optimization techniques, Univ. of California Press, Berkeley and Los Angeles, 1963, 251–278.

    Google Scholar 

  16. [16]

    D. Lubell, A short proof of Sperner’s lemma,J. Combin. Theory1 (1966), 299.

    MathSciNet  Google Scholar 

  17. [17]

    L. D. Meshalkin, A generalization of Sperner’s Theorem on the set of subsets of a finite set.Teor. Verojatnost. i Primenen.8 (1963), 219–220 (in Russian).

    MathSciNet  Google Scholar 

  18. [18]

    D. K. Ray-Chaudhuri andR. M. Wilson, Ont-designs,Osaku J. Math.12 (1975), 737–744.

    MATH  MathSciNet  Google Scholar 

  19. [19]

    K. Yamamoto, Logarithmic order of free distributive lattices,J. Math. Soc. Japan6 (1954), 343–353.

    MATH  MathSciNet  Article  Google Scholar 

Download references

Author information

Affiliations

Authors

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Gronau, H.D.O.F. On sperner families in which nok sets have an empty intersection III. Combinatorica 2, 25–36 (1982). https://doi.org/10.1007/BF02579279

Download citation

AMS subject classification (1980)

  • 05 C 65
  • 05 C 35