Let $P$ be a finite projective space of order $q$ and dimension $d$. I am interested in finding the least $k$ such that for any set $S$ of $k$ points of $P$, and for any set $S'$ of $k$ hyperplanes of $P$, some point of $S$ is incident to some hyperplane of $S'$.
I believe I can prove $k=O( q^{\frac{d+1}2})$ fairly easily but I guess that this is far from optimal. Do you think a better upper bound can be obtained, for instance $q^{O(1)}$? Do you see any non-trivial lower bound?