  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 $\chi_m<(1+2/\sqrt3)^2\cdot m+3.501$
holds.
