On sets of points with few odd secants

TL;DR

This paper establishes a lower bound on the number of lines with an odd number of points in a specific point set within a finite projective plane, revealing structural properties of such configurations.

Contribution

It proves a new lower bound on the number of odd secants for sets of size q+2 in projective planes over finite fields with odd q.

Findings

Sets of size q+2 have at least 2q - c odd secants.

The bound is tight up to a constant c.

Provides insight into the combinatorial structure of point sets in finite projective planes.

Abstract

We prove that, for $q$ odd, a set of $q+2$ points in the projective plane over the field with $q$ elements has at least $2q-c$ odd secants, where $c$ is a constant and an odd secant is a line incident with an odd number of points of the set.