Set systems with three intersections


LetX be a finite set ofn elements and ℱ a family of 4a+5-element subsets,a≧6. Suppose that all the pairwise intersections of members of ℱ have cardinality 0,a or 2a+1. We show thatc 1 n 4/3<max |F|<c 2 n 4/3 for some positivec i’s. This answers a question of P. Frankl.

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  1. [1]

    L. Babai andP. Frankl, Note on set intersections,J. Combin. Theory A,28 (1980), 103–105.

    MATH  Article  MathSciNet  Google Scholar 

  2. [2]

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

    Google Scholar 

  3. [3]

    P. Erdős andD. J. Kleitman, On coloring graphs to maximize the proportion of multicoloredk-edges,J. Combinatorial Th.,5 (1968), 164–169.

    Google Scholar 

  4. [4]

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

    Article  MathSciNet  Google Scholar 

  5. [5]

    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 

  6. [6]

    P. Frankl, Families of finite sets with three intersections,Combinatorica,4 (1984), 141–148.

    MATH  Article  MathSciNet  Google Scholar 

  7. [7]

    Z. Füredi, On finite set-systems whose every intersection is a kernel of a star,Discrete Math. 47 (1983), 129–132.

    MATH  Article  MathSciNet  Google Scholar 

  8. [8]

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

    MATH  MathSciNet  Google Scholar 

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Füredi, Z. Set systems with three intersections. Combinatorica 5, 27–31 (1985).

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AMS subject classification (1980)

  • 05 C 35
  • 05 C 65