On the chromatic number of intersection graphs of convex sets in the plane

Let G be the intersection graph of a finite family of convex sets obtained by translations of a fixed convex set in the plane. We show that every such graph with clique number k is (3k - 3)-degenerate. This bound is sharp. As a consequence, we derive that G is (3k - 2)-colorable. We show also that the chromatic number of every intersection graph H of a family of homothetic copies of a fixed convex set in the plane with clique number k is at most 6k - 6.