  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.
