Covering the nn-space by convex bodies and its chromatic number

Rogers [A note on coverings, Matematika 4 (1957) 1–6] proved, for a given closed convex body CC in nn-dimensional Euclidean space RnRn, the existence of a covering for RnRn by translates of CC with density cnlnncnlnn for an absolute constant cc. A few years later, Erdős and Rogers [Covering space with convex bodies, Acta Arith. 7 (1962) 281–285] obtained the existence of such a covering having not only low-density cnlnncnlnn but also low multiplicity c′nlnnc′nlnn for an absolute constant c′c′. In this paper, we give a simple proof of Erdős and Rogers’ theorem using the Lovász Local Lemma. Furthermore, we apply the result to the chromatic number of the unit-distance graph under ℓpℓp-norm.