A Pseudoline Counterexample to the Strong Dirac Conjecture

TL;DR

This paper constructs an infinite family of pseudoline arrangements that disprove the Strong Dirac Conjecture by showing no pseudoline is incident to more than 4n/9 intersection points, highlighting differences from line arrangements.

Contribution

It provides a counterexample to the Strong Dirac Conjecture for pseudolines and explores the structural differences in incidences between points and pseudolines versus lines.

Findings

Counterexample disproves the Strong Dirac Conjecture for pseudolines

Shows pseudoline arrangements can have limited incidences per pseudoline

Raises open problems on incidence structures in pseudoline arrangements

Abstract

We demonstrate an infinite family of pseudoline arrangements, in which an arrangement of n pseudolines has no member incident to more than 4n/9 points of intersection. This shows the "Strong Dirac" conjecture to be false for pseudolines. We also raise a number of open problems relating to possible differences between the structure of incidences between points and lines versus the structure of incidences between points and pseudolines.