Avoiding secants of given size in finite projective planes

TL;DR

This paper investigates the possible sizes of point sets in finite projective planes that avoid intersecting lines at a specific number of points, providing existence results for various configurations and employing polynomial techniques.

Contribution

It establishes the existence of point sets with prescribed intersection properties in finite projective planes, expanding understanding of combinatorial configurations.

Findings

All or almost all sizes are achievable for point sets avoiding a fixed intersection size, except near extremal values.

Existence of point sets with arbitrary prescribed line intersection counts is proven using polynomial methods.

Results contribute to the theory of blocking sets and related combinatorial structures in finite geometry.

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 paper we show a series of results which point out the existence of all or almost all possible values $m\in [0, q^2+q+1]$ for $|S|=m$, provided that $k$ is not close to the extremal values $0$ or $q+1$. Moreover, using polynomial techniques we show the existence of a point set $S$ with the following property: for every prescribed list of numbers $t_1, \ldots t_{q^2+q+1}$, $|S\cap \ell_i|\neq t_i$ holds for the $i$th line $\ell_i$, $\forall i \in \{1, 2, \ldots, q^2+q+1\}$.