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

TL;DR

This paper establishes an upper bound on the multi-fold chromatic number of the plane, showing it grows linearly with the number of colors per point, with a specific constant factor.

Contribution

The paper provides a new upper bound for the multi-fold chromatic number of the plane, improving understanding of coloring constraints for Euclidean space.

Findings

Upper bound for $oldsymbol{ ext{multi-fold chromatic number}}$ established as $oldsymbol{igl(1+2/\sqrt{3}igr)^2 imes m + 3.501$.

The bound applies for all positive integers $oldsymbol{m}$.

The result advances theoretical limits in geometric graph coloring problems.

Abstract

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.