  A well known observation of Lov\'asz is that if a hypergraph is not
$2$-colorable, then at least one pair of its edges intersect at a single
vertex. %This very simple criterion turned out to be extremly useful . In this
short paper we consider the quantitative version of Lov\'asz's criterion. That
is, we ask how many pairs of edges intersecting at a single vertex, should
belong to a non $2$-colorable $n$-uniform hypergraph? Our main result is an
{\em exact} answer to this question, which further characterizes all the
extremal hypergraphs. The proof combines Bollob\'as's two families theorem with
Pluhar's randomized coloring algorithm.
