Coloring the complements of intersection graphs of geometric figures

Let G¯ be the complement of the intersection graph G   of a family of translations of a compact convex figure in RnRn. When n=2n=2, we show that χ(G¯)⩽min{3α(G)-2,6γ(G)}, where γ(G)γ(G) is the size of the minimum dominating set of G  . The bound on χ(G¯)⩽6γ(G) is sharp. For higher dimension we show that χ(G¯)⩽⌈2(n2-n+1)1/2⌉n-1⌈(n2-n+1)1/2⌉(α(G)-1)+1, for n⩾3n⩾3. We also study the chromatic number of the complement of the intersection graph of homothetic copies of a fixed convex body in RnRn.