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

TL;DR

This paper constructs a counterexample to Gr"unbaum's conjecture regarding transversals in collections of convex sets with the (4,3)-property, and also provides a bound when two disjoint compact sets are present.

Contribution

It provides the first known counterexample to Gr"unbaum's conjecture and establishes a new bound for transversals when two disjoint compact sets exist.

Findings

Counterexample disproves Gr"unbaum's conjecture.

Existence of a transversal of size at most 13 with two disjoint compact sets.

Counterexample challenges assumptions in geometric combinatorics.

Abstract

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.