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Designs, Codes and Cryptography

, Volume 37, Issue 1, pp 151–167 | Cite as

Forbidden (0,1)-vectors in Hyperplanes of \(\mathbb{R}^{n}\): The unrestricted case

  • R. AhlswedeEmail author
  • H. Aydinian
  • L. H. Khachatrian
Article

Abstract

In this paper, we continue our investigation on “Extremal problems under dimension constraints” introduced [1]. The general problem we deal with in this paper can be formulated as follows. Let \(\mathbb{U}\) be an affine plane of dimension k in \(\mathbb{R}^{n}\). Given \(F \subset E(n) {\buildrel = \over \Delta} \{0, 1\}^{n} \subset \mathbb{R}^{n}\) determine or estimate \(\max \left\{|{\cal U} \cap E(n)|: {\cal U} \cap F = {\O}\right\}\).

Here we consider and solve the problem in the special case where \({\cal U}\) is a hyperplane in \(\mathbb{R}^{n}\) and the “forbidden set” \(F = E(n,k) {\buildrel = \over \Delta} \left\{x^{n} \in E(n): x^{n} \hbox{has} k \hbox{ones}\right\}\). The same problem is considered for the case, where \(\mathbb{U}\) is a hyperplane passing through the origin, which surprisingly turns out to be more difficult. For this case we have only partial results.

Keywords

combinatorial extremal theory (0, 1)-vectors dimension constraints forbidden weights 

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References

  1. Ahlswede R., Aydinian H., Khachatrian L.H. (2003). Extremal problems under dimension constraints, Discrete Mathematics , Special issue: EuroComb’01. J. Nesetril, M. Noy and O. Serra, (eds.), Vol. 273, No. 1–3 pp. 9–21Google Scholar
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Copyright information

© Springer Science+Business Media, Inc. 2005

Authors and Affiliations

  1. 1.Fakultät für MathematikUniversität BielefeldBielefeldGermany

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