On the chromatic number of RnRn with an arbitrary norm

In this work we study the Nelson–Erdös–Hadwiger problem on coloring metric spaces. Let χ(RKn) be the chromatic number of the space RnRn with an arbitrary norm determined by a centrally symmetric convex body KK. Füredi and Kang (2008) [5] proved that χ(RKn)≤(5+o(1))n. We improve this bound to χ(RKn)≤(4+o(1))n in the general case, and further improvements are obtained in the case of lplp-norms.