In this paper following theorem is proved:
If $V$ is vector space of dimension $n$ over a finite field $F$ of $q$ elements then any subset of $V$ which meets every hyperplane of $V$ contains at least $n(q-1)+1$ points.
Does anyone know results if we take cosets of codimention $2$ instaed of hyperplanes (coset of codimention $1$)? and $q=2$. So the question will be:
What is minimal set in vector space of dimension $n$ over finite field of $2$ elements, that meets every coset of codimention $2$?
I am going to use it for this problem Find minimal set of cosets $C$, so that each $2$ vectors in $A_n$ are in one coset in $C$. In fact these problems are equivalent.