A Note on the Weak Dirac Conjecture

We show that every set P of n non-collinear points in the plane contains a point incident to at least n 3 + 1 of the lines determined by P. In this note we denote by P a set of non-collinear points in the plane, and by L(P) the set of lines determined by P, where a line that passes through at least two points of P is said to be determined by P. For a point P ∈ P, we denote by d(P) the number of lines of L(P) that are incident to P , called the incident-line-number or multiplicity of P ; see [4] and [14]. Finally, we denote by l r the number of lines that pass through precisely r points of P. Dirac's conjecture is a well-known problem in combinatorial geometry. In 1951, Dirac [5] showed that: Theorem 1. Every set P of n non-collinear points in the plane contains a point incident to at least √ n + 1 lines of L(P).


Introduction
In the note we denote by P a set of non-collinear points in the plane, and L(P) the set of lines determined by P, a line that passing through at least two points in P is said to be determined by P. The system of lines spanned by a finite point set in the plane usually called configuration or arrangement. For a point P of P, we shall denote by d(P ) the number of lines of L(P) which are incident to P , and d(P ) is usually called the incidentline-number or multiplicity of P , see [3] and [10] for details. And, we shall denote by l r the number of r-rich lines, that is they passing through precisely r points of P.
The Dirac's conjecture is a well-known problem in combinatorial geometry. It was proven by Dirac from sixty years earlier. In 1951, Dirac ( [5]) showed that, Theorem 1. Every set P of n non-collinear points in the plane contains a point incident to at least √ n lines of L(P).
Dirac ( [5]) also made the following conjecture and checked the truth for n 14. In the case it is easy to see that this is best-possible, if P is equally distributed on two lines which n is even, the situation is similar when n is odd, then this bound is tight.
Conjecture 2 (Dirac's Conjecture). Every set P of n non-collinear points in the plane contains a point incident to at least ⌊ n 2 ⌋ lines of L(P), where ⌊x⌋ denotes the largest integer not exceeded by x.
This conjecture is certainly false for small n, some counter-examples were found by Grünbaum for relatively small values of n, namely n ∈ {15, 19, 25, 31, 37}, see [7] and [8] the electronic journal of combinatorics 22 (2015), #P00 for details. In 2011, Akiyama, Ito, Kobayashi, and Nakamura ([1]) constructed a set P of non-colinear n points for every integer n 8 except n = 12k + 23, k 0, satisfying d(P ) ⌊ n 2 ⌋ for every point P of P. Then the following conjecture is naturally proposed, see [3] for details.
Conjecture 3 (Strong Dirac Conjecture). Every set P of n non-collinear points in the plane contains a point incident to at least n 2 − c 0 lines of L(P), for some constant c 0 . In 1961, Erdös ([6]) proposed the following weakened conjecture.
Conjecture 4 (Weak Dirac Conjecture). Every set P of n non-collinear points contains a point incident to at least n c 1 lines of L(P), for some constant c 1 . In 1983, the Weak Dirac Conjecture was proved independently by Beck [2] and Szemerédi and Trotter [15] for the case with c 1 unspecified or very large.
In 2012, based on Crossing Lemma, Szemerédi-Trotter Theorem and Hirzebruch's Inequality, Payne and Wood ( [12]) proved the following theorem, There are some results from algebraic geometry involving the system of lines spanned by a finite set in the plane. In 1983, Hirzebruch studied arrangements in the complex projective plane by associating with each arrangement some algebraic surfaces, calculating their Chern numbers, and then applying the Bogomolov-Miyaoka-Yau Inequality. In 1984, Hirzebruch [9] obtained the following result, Theorem 7 (Hirzebruch's Inequality). Let P be a set of n points in the plane with at most n − 3 collinear, then In 2003, the following Hirzebruch-type inequality was announced in the thesis [4] as Lemma 2.2 based on Langer's work [11].
Theorem 8. Let P be a set of n points in the plane with at most ⌊ 2 3 n⌋ collinear, then For the Weak Dirac Conjecture, based on results of [11] and [4], we show that, Theorem 9. Every set P of n non-collinear points in the plane contains a point incident to at least ⌈ n 3 ⌉ + 1 lines of L(P), where ⌈x⌉ denotes the smallest integer greater than or equal to x. Proof. Summing the incident-line-numbers counts each r-rich line r times, since each r-rich line contains r points and contributes to the incident-line-number at every point.
Proof of Theorem 9. Note that if P is non-collinear and contains ⌈ n c 1 ⌉+1 or more collinear points, then Theorem 9 holds, thus we may assume that P does not contain ⌈ n 3 ⌉ + 1 collinear points. According to Theorem 8, we have Applying the Lemma 10, we have, Applying the Lemma 11, we have, P ∈P d(P ) n(n + 3) 3 .
Finally, according to the Pigeonhole Principle [14], this implies that every set P of n points does not contain ⌈ n 3 ⌉ + 1 collinear points in the plane contains a point incident to at least ⌈ n 3 ⌉ + 1 lines of L(P). Therefore every set P of n non-collinear points in the plane contains a point incident to at least ⌈ n 3 ⌉ + 1 lines of L(P).