Mathematics – Combinatorics
Scientific paper
2011-12-31
Mathematics
Combinatorics
4 pages, 1 figure
Scientific paper
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.
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