On a Gallai-type problem for lattices (Q1804756)

The author proves the following Helly-type theorem, which is the planar lattice version of the Gallai problem: If \({\mathcal F}\) is a finite family of convex sets in \(E^ 2\), such that any 3 of them have a lattice point in common, then there exist 2 lattice points, which pin down \({\mathcal F}\) (i.e. each set contains at least one of these lattice points). Neither the `3' nor the `1' can be replaced, i.e. the result is tight.