Profile polytopes of some classes of families

Abstract

The profile vector of a family F of subsets of an n-element set is (f 0,f 1,…,f n ) where f i denotes the number of the i-element members of F. The extreme points of the set of profile vectors for some class of families has long been studied. In this paper we introduce the notion of k-antichainpair families and determine the extreme points of the set of profile vectors of these families, extending results of Engel and P.L. Erdős regarding extreme points of the set of profile vectors of intersecting, co-intersecting Sperner families. Using this result we determine the extreme points of the set of profile vectors for some other classes of families, including complement-free k-Sperner families and self-complementary k-Sperner families. We determine the maximum cardinality of intersecting k-Sperner families, generalizing a classical result of Milner from k = 1.

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References

  1. [1]

    A. Bernáth, D. Gerbner: Chain intersecting families, Graphs and Combinatorics 23 (2007), 353–366.

    MathSciNet  MATH  Article  Google Scholar 

  2. [2]

    K. Engel: Sperner Theory, Encyclopedia of Mathematics and its Applications, 65. Cambridge University Press, Cambridge (1997)

    Google Scholar 

  3. [3]

    K. Engel, P. L. Erdős: Sperner families satisfying additional conditions and their convex hulls, Graphs Combin. 5 (1989), 47–56.

    MathSciNet  MATH  Article  Google Scholar 

  4. [4]

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

    MathSciNet  Article  Google Scholar 

  5. [5]

    P. L. Erdős, P. Frankl, G. O. H. Katona: Extremal hypergraph problems and convex hulls, Combinatorica 5 (1985), 11–26.

    MathSciNet  Article  Google Scholar 

  6. [6]

    D. Gerbner, B. Patkós: l-chain profile vectors, SIAM Journal on Discrete Mathematics 22 (2008), 185–193.

    MathSciNet  MATH  Article  Google Scholar 

  7. [7]

    P. Frankl, G. O. H. Katona: Polytopes determined by hypergraph classes, European J. Combin. 63 (1985), 233–243.

    MathSciNet  Google Scholar 

  8. [8]

    G. O. H. Katona: A Simple proof of the Erdős-Chao Ko-Rado theorem, J. Comb. Theory (B) 13 (1972), 183–184.

    MathSciNet  MATH  Article  Google Scholar 

  9. [9]

    E. C. Milner: A combinatorial theorem on systems of sets, J. London Math. Soc. 43 (1968), 204–206.

    MathSciNet  MATH  Article  Google Scholar 

Download references

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Correspondence to Dániel Gerbner.

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Research supported by Hungarian Science Foundation EuroGIGA Grant OTKA NN 102029

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Gerbner, D. Profile polytopes of some classes of families. Combinatorica 33, 199–216 (2013). https://doi.org/10.1007/s00493-013-2917-y

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Mathematics Subject Classification (2000)

  • 05D05