Forbidden Intersection Patterns in the Families of Subsets (Introducing a Method)

  • Gyula O. H. Katona
Part of the Bolyai Society Mathematical Studies book series (BSMS, volume 17)


Let [n] = 1,2,..., n be a finite set, \( \mathcal{F} \subset 2^{[n]} \) ⊂ 2[n] a family of its subsets. In the present paper max \( \left| \mathcal{F} \right| \) will be investigated under certain conditions on the family \( \mathcal{F} \). The well-known Sperner theorem ([14]) was the first such result.


Lower Estimate Intersection Pattern Boolean Lattice Label Path Constant Weight Code 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    Noga Alon and Joel H. Spencer, The Probabilistic Method, John Wiley & Sons (1992).Google Scholar
  2. [2]
    B. Bollobás, On generalized graphs, Ada. Math. Acad. Sci. Hungar., 16 (1965), 447–452.zbMATHCrossRefGoogle Scholar
  3. [3]
    Annalisa de Bonis, Gyula O.H. Katona, Forbidden r-forks, to appear in Order.Google Scholar
  4. [4]
    Annalisa De Bonis, Gyula O.H. Katona, Konrad J. Swanepoel, Largest family without ABCD, J. Combin. Theory Ser. A, 111, (2005), 331–336.zbMATHCrossRefMathSciNetGoogle Scholar
  5. [5]
    R. C. Bose and T. R. N. Rao, On the theory of unidirectional error correcting codes, Technical report, SC-7817, Sept. 1978, Dept. Comp. Sci., Southern Methodist University, Dallas, TX.Google Scholar
  6. [6]
    Teena Carroll, Gyula O.H. Katona, Largest family without an induced AB, AC, submitted.Google Scholar
  7. [7]
    Konrad Engel, Sperner Theory, Encyclopedia of Mathematics and its Applications 65 (1997), Cambridge University Press.Google Scholar
  8. [8]
    P. Erdös, On a lemma of Littlewood and Offord, Bull. Amer. Math. Soc, 51 (1945) 898–902.zbMATHCrossRefMathSciNetGoogle Scholar
  9. [9]
    R. L. Graham and H. J. A. Sloane, Lower bounds for constant weight codes, IEEE IT, 26, 37–43.Google Scholar
  10. [10]
    Jerrold R. Griggs and Gyula O.H. Katona, No four sets forming an N, to appear in J. Combin Theory.Google Scholar
  11. [11]
    G.O.H. Katona and T. G. Tarján, Extremal problems with excluded subgraphs in the n-cube, Lecture Notes in Math., 1018, 84–93.Google Scholar
  12. [12]
    D. Lubell, A short proof of Sperner’s lemma, J. Combin. Theory, 1 (1966), 299.CrossRefMathSciNetGoogle Scholar
  13. [13]
    L.D. Meshalkin, A generalization of Sperner’s theorem on the number of subsets of a finite set, Teor. Verojatnost. i Primen., 8 (1963), 219–220 (in Russian with German summary).Google Scholar
  14. [14]
    E. Sperner, Ein Satz über Untermegen einer endlichen Menge, Math. Z., 27 (1928), 544–548.zbMATHCrossRefMathSciNetGoogle Scholar
  15. [15]
    Hai Tran Thanh, An extremal problem with excluded subposets in the Boolean lattice, Order, 15 (1998), 51–57.zbMATHCrossRefMathSciNetGoogle Scholar
  16. [16]
    R. R. Varshamov and G. M. Tenengolts, A code which corrects a single asymetric error (in Russian), Avtom. Telemech., 26(2) (1965), 288–292.Google Scholar
  17. [17]
    K. Yamamoto, Logarithmic order of free distributive lattices, J. Math. Soc. Japan, 6 (1954), 347–357.CrossRefGoogle Scholar

Copyright information

© János Bolyai Mathematical Society and Springer-Verlag 2008

Authors and Affiliations

  • Gyula O. H. Katona
    • 1
  1. 1.Rényi InstituteBudapestHungary

Personalised recommendations