  Erd\H{o}s and Lov\'asz noticed that an $r$-uniform intersecting hypergraph
$H$ with maximal covering number, that is $\tau(H)=r$, must have at least
$\frac{8}{3}r-3$ edges. There has been no improvement on this lower bound for
45 years. We try to understand the reason by studying some small cases to see
whether the truth lies very close to this simple bound. Let $q(r)$ denote the
minimum number of edges in an intersecting $r$-uniform hypergraph. It was known
that $q(3)=6$ and $q(4)=9$. We obtain the following new results: The extremal
example for uniformity 4 is unique. Somewhat surprisingly it is not symmetric
by any means. For uniformity 5, $q(5)=13$, and we found 3 examples, none of
them being some known graph. We use both theoretical arguments and computer
searches. In the footsteps of Erd\H{o}s and Lov\'asz, we also consider the
special case, when the hypergraph is part of a finite projective plane. We
determine the exact answer for $r\in \{3,4,5,6\}$. For uniformity 6, there is a
unique extremal example.
  In a related question, we try to find $2$-intersecting $r$-uniform
hypergraphs with maximal covering number, that is $\tau(H)=r-1$. An infinite
family of examples is to take all possible $r$-sets of a $(2r-2)$-vertex set.
There is also a geometric candidate: biplanes. These are symmetric 2-designs
with $\lambda=2$. We determined that only 3 biplanes of the 18 known examples
are extremal.
