  A collection of sets $\Fscr$ has the $(p,q)$-property if out of every $p$
elements of $\Fscr$ there are $q$ that have a point in common. A transversal of
a collection of sets $\Fscr$ is a set $A$ that intersects every member of
$\Fscr$. Gr\"unbaum conjectured that every family $\Fscr$ of closed, convex
sets in the plane with the $(4,3)$-property and at least two elements that are
compact has a transversal of bounded cardinality. Here we construct a
counterexample to his conjecture. On the positive side, we also show that if
such a collection $\Fscr$ contains two {\em disjoint} compacta then there is a
transveral of cardinality at most 13.
