An Erdös-Ko-Rado theorem for the subcubes of a cube


LetP be that partially ordered set whose elements are vectors x=(x 1, ...,x n ) withx i ε {0, ...,k} (i=1, ...,n) and in which the order is given byxy iffx i =y i orx i =0 for alli. LetN i (P)={x εP : |{j:x j ≠ 0}|=i}. A subsetF ofP is called an Erdös-Ko-Rado family, if for allx, y εF it holdsxy, x ≯ y, and there exists az εN 1(P) such thatzx andzy. Let ℓ be the set of all vectorsf=(f 0, ...,f n ) for which there is an Erdös-Ko-Rado familyF inP such that |N i (P) ∩F|=f i (i=0, ...,n) and let 〈ℓ〉 be its convex closure in the (n+1)-dimensional Euclidean space. It is proved that fork≧2 (0, ..., 0) and\(\left( {0,...,0,\overbrace {i - component}^{\left( {\begin{array}{*{20}c} {n - 1} \\ {i - 1} \\ \end{array} } \right)}k^{i - 1} ,0,...,0} \right)\) (i=1, ...,n) are the vertices of 〈ℓ〉.

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  1. [1]

    K. Engel,Maximale h-Familien in endlichen Ordnungen, Hansel-Ordnungen und monotone Funktionen, Dissertation A, Wilh.-Pieck-Univ., Rostock, 1981.

    Google Scholar 

  2. [2]

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

    MATH  Article  Google Scholar 

  3. [3]

    P. L. Erdös, P. Frankl andG. O. H. Katona, Intersecting Sperner families and their convex hulls,Combinatorica 4 (1984), 21–34.

    MATH  MathSciNet  Google Scholar 

  4. [4]

    C. Greene, G. O. H. Katona andD. J. Kleitman, Extensions of the Erdös—Ko—Rado Theorem,Studies in Appl. Math. 55 (1976), 1–8.

    MathSciNet  Google Scholar 

  5. [5]

    H.-D. O. F. Gronau,Zur Theorie der extremalen Familien von Teilmengen einer endlichen Menge, Dissertation B, Wilh.-Pieck-Univ., Rostock, 1982.

    Google Scholar 

  6. [6]

    W. N. Hsieh, Families of intersecting finite vector spaces,J. Combin. Theory Ser. A 18 (1975), 252–261.

    MATH  Article  MathSciNet  Google Scholar 

  7. [7]

    K. Leeb, Salami-Taktik beim Quader-Packen,Arbeitsberichte des Instituts für Mathematische Maschinen und Datenverarbeitung (Informatik), Friedrich Alexander Universität Erlangen Nürnberg11, Nr. 5 (1978), 1–15.

    MathSciNet  Google Scholar 

  8. [8]

    N. Metropolis andG.-C. Rota, Combinatorial structure of the faces of then-cube,SIAM J. Appl. Math. 35 (1978), 689–694.

    MATH  Article  MathSciNet  Google Scholar 

  9. [9]

    A. Moon, An analogue of the Erdös—Ko—Rado Theorem for the Hamming SchemesH(n, q),J. Combin. Theory Ser. A 32 (1982), 386–390.

    MATH  Article  MathSciNet  Google Scholar 

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Engel, K. An Erdös-Ko-Rado theorem for the subcubes of a cube. Combinatorica 4, 133–140 (1984).

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AMS subject classification (1980)

  • 05 A 99
  • 06 A 10