Covering the image-space by convex bodies and its chromatic number Original Research Article

Rogers [A note on coverings, Matematika 4 (1957) 1–6] proved, for a given closed convex body image in image-dimensional Euclidean space image, the existence of a covering for image by translates of image with density image for an absolute constant image. 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 image but also low multiplicity image for an absolute constant image. 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 image-norm.