Line arrangements with the maximal number of triple points
  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. 
Marcin Dumnicki, Lucja Farnik, Agata Glowka, Magdalena   Lampa-Baczynska, Grzegorz Malara, Tomasz Szemberg, Justyna Szpond, ...
math.CO math.AG