On the chromatic number of with an arbitrary norm

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