A new generalization of the Erdős-Ko-Rado theorem


Let ℱ be a family ofk-subsets of ann-set. Lets be a fixed integer satisfyingks≦3k. Suppose that forF 1,F 2,F 3 ∈ ℱ |F 1F 2F 3|≦s impliesF 1F 2F 3 ≠ 0. Katona asked what is the maximum cardinality,f(n, k, s) of such a system. The Erdős-Ko-Rado theorem impliesf(n, k, s)=\(\left( {_{k - 1}^{n - 1} } \right)\) fors=3k andn≧2k. In this paper we show thatf(n, k, s)=\(\left( {_{k - 1}^{n - 1} } \right)\) holds forn>n 0(k) if and only ifs≧2k.

Equality holds only if every member of ℱ contains a fixed element of the underlying set.

Further we solve the problem fork=3,s=5,n≧3000. This result sharpens a theorem of Bollobás.

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  1. [1]

    B. Bollobás, Three-graphs without two triples whose symmetric difference is contained in a third,Discrete Math. 8 (1974) 21–24.

    MATH  Article  MathSciNet  Google Scholar 

  2. [2]

    P. Erdős, Problems and results in graph theory and combinatorial analysis,Proc. Fifth British Comb. Conf. 1975, Aberdeen 1975 (Utilitas Math. Winnipeg (1976), 169–172.

  3. [3]

    P. Erdős, On the number of complete subgraphs contained in a certain graphs,Publ. Math. Inst. of the Hungar. Acad. Sci. (Ser A) 7 (1962) 459–464.

    Google Scholar 

  4. [4]

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

    Article  MathSciNet  Google Scholar 

  5. [5]

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

    MATH  Article  MathSciNet  Google Scholar 

  6. [6]

    P. Frankl, On a problem of Chvátal and Erdős,J. Combinatorial Th. A, to appear.

  7. [7]

    P. Frankl, On families of finite sets no two of which intersect in a singleton,Bull. Austral. Math. Soc. 17 (1977) 125–134.

    MATH  MathSciNet  Article  Google Scholar 

  8. [8]

    P. Frankl, A general intersection theorem for finite sets,Proc. of French-Canadian Combinatorial Coll., Montreal 1979, Annals of Discrete Math. 8 (1980), to appear.

  9. [9]

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

    MATH  Article  MathSciNet  Google Scholar 

  10. [10]

    P. Turán, An extremal problem in graph theory,Mat. Fiz. Lapok 48 (1941) 436–452 (in Hungarian).

    MATH  MathSciNet  Google Scholar 

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Dedicated to Paul Erdős on his seventieth birthday

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Frankl, P., Füredi, Z. A new generalization of the Erdős-Ko-Rado theorem. Combinatorica 3, 341–349 (1983). https://doi.org/10.1007/BF02579190

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AMS subject classification (1980)

  • 05 C 35
  • 05 B 30