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.
Highlights
A central problem of discrete geometry is to elucidate the structure of incidences between points and lines
Until the recent explosion of applications of polynomial methods to problems in incidence geometry ([7, 16, 19]), the tools most successfully applied to questions about incidences between points and lines could be immediately applied to prove equivalent results for incidences between points and pseudolines
The main interest of the pseudoline arrangement presented here is that it shows that a natural conjecture that is widely believed to be true for straight lines is definitely false for pseudolines
Summary
Submitted: Feb 14, 2012; Accepted: Apr 23, 2014; Published: May 13, 2014 Mathematics Subject Classification: 52C10
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