  In this paper we study some Erdos type problems in discrete geometry. Our
main result is that we show that there is a planar point set of n points such
that no four are collinear but no matter how we choose a subset of size
$n^{5/6+o(1)} $ it contains a collinear triple. Another application studies
epsilon-nets in a point-line system in the plane.
  We prove the existence of some geometric constructions with a new tool, the
so-called Hypergraph Container Method.
