  We consider a question raised by Rudnev: given four pencils of $n$ concurrent
lines in $\mathbb R^2$, with the four centres of the pencils non-collinear,
what is the maximum possible size of the set of points where four lines meet?
Our main result states that the number of such points is $O(n^{11/6})$,
improving a result of Chang and Solymosi.
  We also consider constructions for this problem. Alon, Ruzsa and Solymosi
constructed an arrangement of four non-collinear $n$-pencils which determine
$\Omega(n^{3/2})$ four-rich points. We give a construction to show that this is
not tight, improving this lower bound by a logarithmic factor. We also give a
construction of a set of $m$ $n$-pencils, whose centres are in general
position, that determine $\Omega_m(n^{3/2})$ $m$-rich points.
