A new progress on Weak Dirac conjecture

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.