On upper bounds for the multi-fold chromatic numbers of the plane

The multi-fold chromatic number of the plane ${\chi }_m$ is the smallest number of colors $k$, sufficient to color each point of the Euclidean plane in exactly $m$ colors, so that for any pair of points at a unit distance from each other, two corresponding $m$-subsets of $k$-set do not contain any common color. We consider upper bounds for $m$-fold chromatic numbers of the plane. Our main result is that for any $m$ the inequality ${\chi }_m<(1+2\mathrm{/}\sqrt{3}{)}^2\cdot m+3.501$ holds.