A new progress on Weak Dirac conjecture

TL;DR

This paper improves the lower bound on the number of lines determined by a non-collinear point set in the plane, refining previous results and discussing related theorems in combinatorial geometry.

Contribution

It refines Payne-Wood's bound to show each non-collinear set contains a point in at least n/26+2 lines, advancing understanding of the weak Dirac conjecture.

Findings

Improved lower bound from n/37 to n/26+2 lines

Established relations with Beck's theorem on collinear points

Enhanced understanding of point-line incidences in planar geometry

Abstract

In 2014, Payne-Wood proved that every non-collinear set $P$ of $n$ points in the Euclidean plane contains a point in at least $\dfrac{n}{37}$ lines determined by $P.$ This is a remarkable answer for the conjecture, which was proposed by Erd\H{o}s, that every non-collinear set $P$ of $n$ points contains a point in at least $\dfrac{n}{c_1}$ lines determined by $P$, for some constant $c_1.$ In this article, we refine the result of Payne-Wood to give that every non-collinear set $P$ of $n$ points contains a point in at least $\dfrac{n}{26}+2$ lines determined by $P$ . Moreover, we also discuss some relations on theorem Beck that every set $P$ of $n$ points with at most $l$ collinear determines at least $\dfrac{1}{61}n(n-l)$ lines.