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

• Authors: R. Ahlswede;H. Aydinian;L. H. Khachatrian
• Venue: Designs, Codes and Cryptography
• Year: 2005

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.