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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 ofa collection of sets $\Fscr$ is a set $A$ that intersects every member of$\Fscr$. Gr\"unbaum conjectured that every family $\Fscr$ of closed, convexsets in the plane with the $(4,3)$-property and at least two elements that arecompact has a transversal of bounded cardinality. Here we construct acounterexample to his conjecture. On the positive side, we also show that ifsuch a collection $\Fscr$ contains two {\em disjoint} compacta then there is atransveral of cardinality at most 13.

A counterexample to conjecture 18.5 in "Geometric Etudes in Combinatorial Mathematics", second edition