Using block designs in crossing number bounds

The crossing number CR(G) of a graph G=(V,E) is the smallest number of edge crossings over all drawings of G in the plane. For any k >= 1, the k-planar crossing number of G,CRk(G), is defined as the minimum of CR(G1)+CR(G2)+MIDLINE HORIZONTAL ELLIPSIS+CR(Gk) over all graphs G1,G2, horizontal ellipsis ,Gk with ?i=1kGi=G. Pach et al [Comput. Geom.: Theory Appl. 68 (2018), pp. 2-6] showed that for every k >= 1, we have CRk(G)<=(2/k2-1/k3)CR(G) and that this bound does not remain true if we replace the constant 2/k2-1/k3 by any number smaller than 1/k2. We improve the upper bound to (1/k2)(1+o(1)) as k ->infinity. For the class of bipartite graphs, we show that the best constant is exactly 1/k2 for every k. The results extend to the rectilinear variant of the k-planar crossing number.