Let $\mathbb F\_q$ be the finite field with $q$ elements. Suppose $V$ is a linear space of dimension $n$ over $\mathbb F\_q$, and $r<n$. What is the maximal $k$ such that for arbitrary $k$ subspaces $W\_1,W\_2,\dots,W\_k$ of $V$ of dimension $r$, there always exists a subspace $U$ of $V$ of dimension $n-r$ which satisfies $U\cap W\_i=\{0\}$ for each $W\_i$? Thank you.


PS. The choice of $W\_i$ is arbitrary, and I didn't say it clearly in the former version, sorry. 


PS. I guess the number might be something like $q^{n-r}$, but I don't know how to prove it.


PS. The answer is $k=q$. 



What is the maximal number of sub spaces of a fixed dimension such that there is another sub space which intersects them are all null
I think we have $k = q$ (for all $r$). 


First, suppose $r = n - 1$. The vector space $V$ can be covered by $q + 1$ codimension-one hyperplanes $W\_1,W\_2, \ldots,W\_{q+1}$ (but no fewer). (If we think of the projective space $[V]$, the most efficient covering is by taking the hyperplane at infinity, plus a set of $q$ parallel hyperplanes in affine space.) Then every $1$-dimensional subspace intersects (in fact, is contained in) some $W\_i$. So $k = q$ when $r = n-1$.


To prove $k \le q$ in general, fix a subspace $Z$ of $V$ that has dimension $r + 1$, and use the previous case to choose $q + 1$ subspaces $W\_1,\ldots,W\_{q+1}$ of dimension $r$ in $Z$, such that every $1$-dimensional subspace of $Z$ is contained in some $W\_i$. Since every subspace $U$ of dimension $n - r$ in $V$ contains a subspace of $Z$ that has dimension (at least) $1$, we conclude that $U$ intersects some $W\_i$. So $k < q + 1$, as desired.


Now we prove the matching lower bound: $k \ge q$. Let $W\_1,W\_2, \ldots,W\_q$ be any $r$-dimensional subspaces of $V$. Since (as mentioned above), it takes $q + 1$ hyperplanes to cover all of $V$, we know there is some $1$-dimensional subspace $L$ that is not contained in $\bigcup\_i W\_i$. Let $\overline{V} = V/L$. By induction on $n - r$, there is an $(n-r-1)$-dimensional subspace $\overline{U}$ of $\overline{V}$ that has trivial intersection with every $\overline{W\_i}$. Lift $\overline{U}$ to an $(n-r)$-dimensional subspace of $V$. Then $U$ has trivial intersection with every $W\_i$.


