A Note on Weak Dirac Conjecture

TL;DR

This paper proves that in any set of non-collinear points in the plane, there exists a point incident to at least roughly one-third of the lines determined by these points, advancing understanding of point-line incidences.

Contribution

It establishes a new lower bound on the maximum number of lines incident to a single point in a non-collinear point set, contributing to the weak Dirac conjecture.

Findings

Existence of a point incident to at least rac{n}{3}rac{n}{3}+1 lines

Improved lower bound on point-line incidences in planar point sets

Progress towards the weak Dirac conjecture

Abstract

We show that every set $\mathcal{P}$ of $n$ non-collinear points in the plane contains a point incident to at least $\lceil\frac{n}{3}\rceil+1$ of the lines determined by $\mathcal{P}$.