Let $S$ be a finite set of non-zero points from a vector space $V=F^n$. I now want to find a subspace $ U\subseteq V$ that does not contain any (or contains as few as possible) points from $S$.
Specifically, I have the two similar problems:
- Fix the dimension of $U$ as $m$ and minimize the number of points from $S$ still contained in $U$.
- Do not allow any points from $S$ in $U$ and maximize the dimension $m$ of $U$.
If $F=\mathbb{R}$, I believe $\dim(U)$ can be as high as $n-1$. However, for a finite field $F$, both problems look to me as (possibly hard) integer-programming tasks. Even if there is no general solution to either of them, I'd like to know if these or related problems have already been studied before and maybe there are some hints for solving them (algorithms, etc.).
This seems to be a generalized version of this question, where the field was infinite: Existence of lines not containing given points in general position