On disjointly representable sets


A system of setsE 1,E 2, ...,E kX is said to be disjointly representable if there existx 1,x 2, ...,x k teX such thatx i teE j i=j. Letf(r, k) denote the maximal size of anr-uniform set-system containing nok disjointly representable members. In the first section the exact value off(r, 3) is determined and (asymptotically sharp) bounds onf(r, k),k>3 are established. The last two sections contain some generalizations, in particular we prove an analogue of Sauer’ theorem [16] for uniform set-systems.

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  1. [1]

    C. Berge,Graphs and Hypergraphs, North-Holland, 1973.

  2. [2]

    B. Bollobás, On generalized graphs,Acta Math. Hung. 16 (1965), 447–452.

    MATH  Article  Google Scholar 

  3. [3]

    P. Erdős, On bipartite subgraphs of a graph (in Hungarian),Matematikai Lapok 18 (1967), 283–288.

    MathSciNet  Google Scholar 

  4. [4]

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

    Google Scholar 

  5. [5]

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

    Article  Google Scholar 

  6. [6]

    P. Erdős andL. Moser, An extremal problem in graph theory,J. Austral. Math. Soc. 11 (1970), 42–47.

    MathSciNet  Article  Google Scholar 

  7. [7]

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

    Article  MathSciNet  Google Scholar 

  8. [8]

    P. Frankl, On the trace of finite sets,J. Combinatorial Th. Ser. A,34 (1983), 41–45.

    MATH  Article  MathSciNet  Google Scholar 

  9. [9]

    P. Frankl andJ. Pach, On the number of sets in a null-t-design,European J. Comb. 4 (1983), 21–33.

    MATH  MathSciNet  Google Scholar 

  10. [10]

    F. Jaeger andC. Payan, Determination du nombre d’aretes d’un hypergraphe τ-critique,C. R. Acad. Sc. Paris 273 (1971), 221–223.

    MATH  MathSciNet  Google Scholar 

  11. [11]

    A. Hajnal, Personal communication.

  12. [12]

    G. O. H. Katona, Solution of a problem of A. Ehrenfeucht and J. Mycielsky,J. Comb. Th. 17 (1974), 265–266.

    MATH  Article  MathSciNet  Google Scholar 

  13. [13]

    G. Katona, T. Nemetz andM. Simonovits, On a graph problem of Turán (in Hungarian),Matematikai Lapok 15 (1964), 228–238.

    MATH  MathSciNet  Google Scholar 

  14. [14]

    L. Lovász, Topological and algebraic methods in graph theory, in:Graph Theory and Related topics (J. A. Bondy and U. S. R. Murty, eds.), Academic Press, New York 1979, 1–14.

    Google Scholar 

  15. [15]

    V. Rödl, Almost Steiner systems always exist,to appear in European J. of Comb.

  16. [16]

    N. Sauer, On the density of families of sets,J. Combinatorial Th. Ser. A,13 (1972), 145–147.

    MATH  Article  MathSciNet  Google Scholar 

  17. [17]

    A. F. Sidorenko, On the Turán numberT(n, 5, 4) and on the number of monochromatic 4-cliques in a two-coloured 3-graph (in Russian),Voprosi Kibernetiki, Komb. Anal. i Teoria Grafov, Nauka, Moscow 1980, 117–124.

    Google Scholar 

  18. [18]

    P. Turán, On the theory of graphs,Coll. Math. 3 (1954), 19–30.

    MATH  Google Scholar 

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Dedicated to Paul Erdős on his seventieth birthday

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Frankl, P., Pach, J. On disjointly representable sets. Combinatorica 4, 39–45 (1984). https://doi.org/10.1007/BF02579155

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AMS subject classification (1980)

  • 05 C 35
  • 05 C 65
  • 05 A 05