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

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.