Decomposition of multiple coverings into many parts

Let m(k) denote the smallest positive integer m such that any m-fold covering of the plane with axis-parallel unit squares splits into at least k coverings. J. Pach [J. Pach, Covering the plane with convex polygons, Discrete and Computational Geometry 1 (1986) 73–81] showed that m(k) exists and gave an exponential upper bound. We show that m(k)=O(k2), and generalize this result to translates of any centrally symmetric convex polygon in the place of squares. From the other direction, we know only that m(k)⩾⌊4k/3⌋−1.