  A family of sets has the $(p, q)$ property if among any $p$ members of it
some $q$ intersect. It is shown that if a finite family of compact convex sets
in $\R^2$ has the $(p+1,2)$ property then it is pierced by $\lfloor \frac{p}{2}
\rfloor +1$ lines. A colorful version of this result is proved as well. As a
corollary, the following is proved: Let $\F$ be a finite family of compact
convex sets in the plane with no isolated sets, and let $\F'$ be the family of
its pairwise intersections. If $\F$ has the $(p+1,2)$ property and $\F'$ has
the $(r+1,2)$ property, then $\F$ is pierced by $(\lfloor \frac{r}{2} \rfloor
^2 +\lfloor\frac{r}{2} \rfloor)p$ points when $r\ge 2$, and by $p$ points
otherwise. The proofs use the topological KKM theorem.
