On a Gallai-type problem for lattices

Motivated by the well-known Helly-theorem [2], Gailai [1] raised the following problem in the Euclidean plane E 2. Let D denote a finite collection of closed disks in E 2 such that every two disks of D intersect. Find the minimum integer n with the property that for an arbitrary D there are n points in E 2 such that every disk of D contains at least one of the points. Independently from each other, Danzer (unpublished) and Stach5 [3] proved that n =< 4 i.e. any D can be pinned down by 4 needles. An analogous problem arises if the needles can be chosen from a rather regular subset of E 2 only. Let L be the lattice of E 2, i.e. the set of points of E 2 which have integer coordinates. It is easy to prove the following Helly-type theorem (see [4]). If ~" is a finite collection of convex sets in E 2 such that any four of the sets of ~" have a lattice point in common, then there exists a lattice point common to every set of.T. Moreover this theorem can be extended to the d-dimensional Euclidean space E d replacing 4 by 2 d. Thus it is very natural to ask the following Gallaitype problem for planar lattices. Let ~" denote a finite collection of convex sets in E 2 such that any three of the sets of ~ have a lattice point in common. Find the least integer n such that for an arbitrary Jr there exist n lattice points (i.e. n needles positioned at the lattice points) with the property that every set of .T contains (i.e. is pinned down) by at least one of the n lattice points (i.e. needles). We prove the following