  A long standing conjecture of Richter and Thomassen states that the total
number of intersection points between any $n$ simple closed Jordan curves in
the plane, so that any pair of them intersect and no three curves pass through
the same point, is at least $(1-o(1))n^2$.
  We confirm the above conjecture in several important cases, including the
case (1) when all curves are convex, and (2) when the family of curves can be
partitioned into two equal classes such that each curve from the first class is
touching every curve from the second class. (Two curves are said to be touching
if they have precisely one point in common, at which they do not properly
cross.)
  An important ingredient of our proofs is the following statement: Let $S$ be
a family of the graphs of $n$ continuous real functions defined on
$\mathbb{R}$, no three of which pass through the same point. If there are $nt$
pairs of touching curves in $S$, then the number of crossing points is
$\Omega(nt\sqrt{\log t/\log\log t})$.
