Line arrangements with the maximal number of triple points

TL;DR

This paper investigates the maximum number of triple intersection points in line arrangements over various fields, establishing existence conditions, disproving certain configurations, and presenting an infinite series with many triple points.

Contribution

It provides precise conditions for the existence of extremal line arrangements over different fields and introduces an infinite series of configurations with many triple points.

Findings

No configuration of 11 lines with 17 triple points exists over any field.

Conditions for the existence of extremal configurations depend on the ground field.

An infinite series of configurations with high triple intersection points is constructed.

Abstract

The purpose of this note is to study configurations of lines in projective planes over arbitrary fields having the maximal number of intersection points where three lines meet. We give precise conditions on ground fields F over which such extremal configurations exist. We show that there does not exist a field admitting a configuration of 11 lines with 17 triple points, even though such a configuration is allowed combinatorially. Finally, we present an infinite series of configurations which have a high number of triple intersection points.