Families of finite sets with three intersections


LetX be a finite set ofn elements and ℓ a family ofk-subsets ofX. Suppose that for a given setL of non-negative integers all the pairwise intersections of members of ℓ have cardinality belonging toL. Letm(n, k, L) denote the maximum possible cardinality of ℓ. This function was investigated by many authors, but to determine its exact value or even its correct order of magnitude appears to be hopeless. In this paper we investigate the case |L|=3. We give necessary and sufficient conditions form(n, k, L)=O(n) andm(n, k, L)≧O(n 2), and show that in some casesm(n, k, L)=O(n 3/2), which is quite surprising.

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  1. [1]

    L. Babai andP. Frankl, On set intersections,J. Comb. Th. A 28 (1980), 103–105.

    MATH  Article  MathSciNet  Google Scholar 

  2. [2]

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

    MATH  Article  MathSciNet  Google Scholar 

  3. [3]

    M. Deza, P. Erdös andN. M. Singhi, Combinatorial problems on subsets and their intersections,Advances in Math. Suppl. Stud. 1 (1978), 259–265.

    Google Scholar 

  4. [4]

    P. Erdös andR. Rado, Intersection theorems for systems of sets,J. London Math. Soc. 35 (1960), 85–90.

    MATH  Article  MathSciNet  Google Scholar 

  5. [5]

    P. Frankl andR. M. Wilson, Intersection theorems with geometric consequences,Combinatorica 1 (1981), 357–368.

    MATH  MathSciNet  Google Scholar 

  6. [6]

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

    MATH  Article  MathSciNet  Google Scholar 

  7. [7]

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

    MATH  MathSciNet  Google Scholar 

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Frankl, P. Families of finite sets with three intersections. Combinatorica 4, 141–148 (1984). https://doi.org/10.1007/BF02579214

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AMS subject classification (1980)

  • 05 A 05
  • 05 C 35
  • 05 C 65