, Volume 25, Issue 3, pp 229–236 | Cite as

Bounds on Maximal Families of Sets Not Containing Three Sets with ABC, AB

  • Teena Carroll
  • Gyula O. H. Katona


Lower and upper estimates are given on the size of a family of subsets of an n-element set containing no three distinct sets satisfying ABC, AB. This is a sharpening of an earlier result where the same question was solved under the condition that there are no three distinct sets such that A ∩ B ⊂ C.


Extremal problem for families Sperner type theorem Forbidden subposet 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Alon, N., Spencer, J.H.: The Probabilistic Method, 2nd edn. Wiley, New York (2000)MATHGoogle Scholar
  2. 2.
    De Bonis, A., Katona, G.O.H.: Largest families without an r-fork. Order 24, 181–191 (2007)MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Erdős, P.: On a lemma of Littlewood and Offord. Bull. Am. Math. Soc. 51, 898–902 (1945)CrossRefGoogle Scholar
  4. 4.
    Griggs, J.R., Katona, G.O.H.: No four subsets forming an N. J. Comb. Theory A 115, 677–685 (2008)MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Graham, R.L., Sloane, N.J.A.: Lower bounds for constant weight codes. IEEE IT 26, 37–43 (1980)MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Katona, G.O.H., Tarján, T.G.: Extremal problems with excluded subgraphs in the n-cube. In: Borowiecki, M., Kennedy, J.W., Sysło M.M. (eds.) Graph Theory, Łagów, 1981. Lecture Notes in Math., vol. 1018, pp. 84–93. Springer, Berlin Heidelberg New York (1983)CrossRefGoogle Scholar
  7. 7.
    Lubell, D.: A short proof of Sperner’s lemma. J. Comb. Theory 1, 299 (1966)CrossRefMathSciNetGoogle Scholar
  8. 8.
    Sperner, E.: Ein Satz über Untermegen einer endlichen Menge. Math. Z. 27, 544–548 (1928)MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Thanh, H.T.: An extremal problem with excluded subposets in the Boolean lattice. Order 15, 51–57 (1998)MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2008

Authors and Affiliations

  1. 1.School of MathematicsGeorgia Institute of TechnologyAtlantaUSA
  2. 2.Rényi InstituteBudapestHungary

Personalised recommendations