A Problem Concerning Nonincident Points and Lines in Projective Planes

TL;DR

This paper investigates the maximum size of point-line configurations in projective planes where no point lies on any of the lines, establishing an upper bound and conditions for equality.

Contribution

It introduces a new upper bound for the largest nonincident point-line sets in projective planes and characterizes cases where this bound is tight.

Findings

Established s <= 1+(q+1)(√q - 1) as the upper bound.

Proved that equality holds when q is an even power of two.

Provided constructions achieving the bound for specific q values.

Abstract

In this paper, we study the problem of finding the largest possible set of s points and s lines in a projective plane of order q, such that that none of the s points lie on any of the s lines. We prove that s <= 1+(q+1)(\sqrt{q}-1). We also show that equality can be attained in this bound whenever q is an even power of two.