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On Dirac and Motzkin problem in discrete geometry

Jan Florek

TL;DR

This work addresses the Dirac–Motzkin incidence problem by proving that any non-collinear point set $\mathcal{X}$ distributed on three lines through a common point $A$ contains an ordinary point, i.e., a point incident with at least $\lceil n/2 \rceil$ lines spanned by $\mathcal{X}$. The authors implement a geometric construction using lines $p,q,r$ through $A$ and six half-lines $p_1,p_2,q_1,q_2,r_1,r_2$, then perform a detailed case analysis on the arrangement of the line $P_1Q_1$ with these half-lines. An induction on $|\mathcal{X}|$ coupled with a deletion argument shows that either $P_1$ or $Q_1$ is ordinary, establishing the theorem. This result adds a robust positive instance to the broader Dirac–Motzkin discourse and clarifies incidence structure for configurations restricted to three lines through a common point, linking to projective-to-Euclidean reductions used in the argument.

Abstract

Dirac and Motzkin conjectured that any set X of $n$ non-collinear points in the plane has an element incident with at least $\lceil \frac{n}{2} \rceil$ lines spanned by X. In this paper we prove that any set X of $n$ non-collinear points in the plane, distributed on three lines passing through a common point, has an element incident with at least $\lceil \frac{n}{2} \rceil$ lines spanned by X.

On Dirac and Motzkin problem in discrete geometry

TL;DR

This work addresses the Dirac–Motzkin incidence problem by proving that any non-collinear point set distributed on three lines through a common point contains an ordinary point, i.e., a point incident with at least lines spanned by . The authors implement a geometric construction using lines through and six half-lines , then perform a detailed case analysis on the arrangement of the line with these half-lines. An induction on coupled with a deletion argument shows that either or is ordinary, establishing the theorem. This result adds a robust positive instance to the broader Dirac–Motzkin discourse and clarifies incidence structure for configurations restricted to three lines through a common point, linking to projective-to-Euclidean reductions used in the argument.

Abstract

Dirac and Motzkin conjectured that any set X of non-collinear points in the plane has an element incident with at least lines spanned by X. In this paper we prove that any set X of non-collinear points in the plane, distributed on three lines passing through a common point, has an element incident with at least lines spanned by X.

Paper Structure

This paper contains 2 sections, 1 theorem, 2 figures.

Table of Contents

  1. Introduction
  2. Main result

Key Result

theorem 1

Any set $\mathcal{X}$ of $n$ non-collinear points in the plane distributed on three lines passing through a point $A$$($$A$ is not required to be in the set $\mathcal{X}$$)$ has an ordinary point.

Figures (2)

  • Figure 1: Akiyama, Ito, Kobayashi and Nakamura construction for $n=17$ points (including $\infty$)
  • Figure 2: Akiyama, Ito, Kobayashi and Nakamura construction for $n=17$ points (including $\infty_1$ and $\infty_2$) distributed, except for $\infty_2$, on three lines passing through $\infty_1$.

Theorems & Definitions (2)

  • theorem 1
  • proof