On the Helly number for hyperplane transversals to unit balls (Q1580738)

The paper concerns the smallest number \(k\) such that for every family of at least \(k\) pairwise disjoint unit balls in the Euclidean \(d\)-space \(\mathbb{R}^d\) the following implication is true: if every \(k\) from the balls are met by a hyperplane, then there is a hyperplane which meets all the balls from the family. This number is called Helly number for hyperplane transversals to disjoint unit balls. The authors consider arbitrary family of \(d+3\) or more unit balls in \(\mathbb{R}^d\) such that no \(d\) of the balls have a transversal of dimension less than \(d-1\). They prove that the Helly number for hyperplane transversals to this family is at least \(d+3\). Moreover, it is proved that for each \(n\geq 6\) there exist \(n\) pairwise disjoint unit disks in the plane such that every 4 are met by a straight line but some 5 are not.