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Forbidden (0,1)-Vectors in Hyperplanes of ℝn: The Restricted Case

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Abstract

In this paper we continue our investigation on “Extremal problems under dimension constraint” introduced in [2].

Let E(n, k) be the set of (0,1)-vectors in ℝn with k one's. Given 1 ≤ m, w ≤ n let X ⊂ E(n, m) satisfy span (X) ∩ E(n, w) = ⊘. How big can |X| be?

This is the main problem studied in this paper. We solve this problem for all parameters 1 ≤ m, w ≤ n and n > n 0(m, w).

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References

  1. R. Ahlswede, H. Aydinian and L. H. Khachatrian, Maximal number of constant weight vertices of the unit n-cube containing in a k-dimensional subspace, submitted to Combinatorica, special issue dedicated to the memory of Paul Erdős.

  2. R. Ahlswede, H. Aydinian and L. H. Khachatrian, Extremal problems under dimension constraint, preprint 00–116, SFB 343, “Diskrete Strukturen in der Mathematik,” Universität Bielefeld (2000), submitted to the Special Volume of Discrete Mathematics devoted to selected papers from EUROCOMB'01.

  3. R. Ahlswede, H. Aydinian and L. H. Khachatrian, Maximal antichains under dimension constraints, preprint 00–116, SFB 343, “Diskrete Strukturen in der Mathematik,” Universität Bielefeld (2000), submitted to the Special Volume of Discrete Mathematics devoted to selected papers from EUROCOMB'01.

  4. R. Ahlswede and L. H. Khachatrian, The complete intersection theorem for systems of finite sets, Europ. J. Comb., Vol. 18 (1997) pp. 125–136.

    Google Scholar 

  5. R. Ahlswede and L. H. Khachatrian, The complete nontrivial-intersection theorem for systems of finite sets, J. Combin. Theory Ser. A, Vol. 76 (1996) pp. 121–138.

    Google Scholar 

  6. B. Bollobas, D. E. Daykin and P. Erdős, Sets of independent edges of a hypergraph, Quart. J. Math., Vol. 21 (1976) pp. 25–32.

    Google Scholar 

  7. K. Engel, Sperner Theory, Cambridge University Press (1997).

  8. P. Erdős, A Problem of Independent r-Tuples, Annales Univ. Budapest, Vol. 8 (1965) pp. 93–95.

    Google Scholar 

  9. P. Frankl, On intersecting families of finite sets, J. Combin. Theory Ser. A, Vol. 24 (1978) pp. 141–161.

    Google Scholar 

  10. P. Frankl and Z. Füredi, Non-trivial intersecting families, J. Combin. Theory Ser. A, Vol. 41 (1986) pp. 150–153.

    Google Scholar 

  11. A. J. W. Hilton and E. C. Milner, Some intersection theorems for systems of finite sets, Quart. J. Math., Vol. 18 (1967) pp. 369–384.

    Google Scholar 

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Authors and Affiliations

  1. Fakultät Mathematik, Universität Bielefeld, Universitätsstr. 25, 33615, Bielefeld, Germany

    R. Ahlswede & L. H. Khachatrian

  2. Fakultät für Mathematik, Universität Bielefeld, Postfach 100131, D-33501, Bielefeld, Germany

    H. Aydinian

Authors
  1. R. Ahlswede
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  2. H. Aydinian
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  3. L. H. Khachatrian
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Ahlswede, R., Aydinian, H. & Khachatrian, L.H. Forbidden (0,1)-Vectors in Hyperplanes of ℝn: The Restricted Case. Designs, Codes and Cryptography 29, 17–28 (2003). https://doi.org/10.1023/A:1024131820350

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