Avoiding Secants of Given Size in Finite Projective Planes

Abstract

Let $q$ be a prime power and $k$ be a natural number. What are the possible cardinalities of point sets $S$ in a projective plane of order $q$, which do not intersect any line at exactly $k$ points? This problem and its variants have been investigated before, in relation with blocking sets, untouchable sets or sets of even type, among others. In this article, we show a series of results which point out the existence of all or almost all possible values $m\in \left[0,q^2+q+1\right]$ for $\mid S\mid =m$, provided that $k$ is not close to the extremal values 0 or $q+1$. Moreover, using polynomial techniques we show that for every prescribed list of numbers $t_1,\text{…}{t}_{q^2+q+1}$, there exists a point set $S$ with the following property: $\mid S\cap {\ell }_i\mid \ne t_i$ holds for the $i$th line ${\ell }_i$, $\forall i\in \left\{1,2,\text{…},q^2+q+1\right\}$.