Hyperplanes not intersecting points on a cube Consider the set of points in $\mathbb{R}^n$ with coordinates in $\{-1, 0, 1\}$. Find a hyperplane passing through the origin that contains no points in the set besides the origin.

This is simple if we don't give any constraints on the hyperplane. Even if define the hyperplane by a normal vector in $\mathbb{Z}^n$, it is simple. However, what if we require every coordinate of the normal vector to be an integer bounded in absolute value by some polynomial in $n$? Does such a hyperplane exist?
No. Let the normal vector be $(a_1,\ldots, a_n)$. If these numbers are $poly(n)$ small integers, then there is an $I$ and $J$ such that $\sum_{i\in I} a_i=\sum_{j\in J} a_j$ by the pigeon hole principle. But then we have $\sum_{i=1}^n \varepsilon_i a_i=0$ for some $\varepsilon \in \{-1,0,1\}^n$.