The crossing number of a graph $G$, $cr(G)$, is the smallest number of edge crossings
over all drawings of the graph in the plane. For any $k\geq 1$, the $k$-planar crossing number of $G$
is defined as the minimum of $cr(G_1)+cr(G_2)+...+cr(G_k)$, where the edges of $G$ are partitioned into
the graphs $G_1,G_2,...,G_k$. What is the smallest positive number $\alpha_k$,
such that $cr_k(G)\leq \alpha_k cr(G)$ holds for every graph $G$? Czabarka, S\'ykora, Sz, and Vr\v to
[{\it RSA} {\bf 33} (2008), 480--496] showed $\alpha_2\leq 3/8$, which is still the best result for $k=2$.
Pach, Sz, T\'oth and T\'oth [{\it Comp. Geom.: Theory and Appl.} {\bf 68} (2018), 2--6] estimated $\alpha_k$ within a multiplicative factor of 2.
Now we show that $\alpha_k=1/k^2 +o(1)$ as $k\rightarrow\infty$, and furthermore,
if we restrict the problem to bipartite graphs, it is {\it exactly} $1/k^2$. The upper bounds come from randomized
algorithms that make use the of the existence of RBIBDs, while the lower bounds utilize the existence of the
midrange crossing constant of Pach, Spencer and T\'oth [{\it DCG} {\bf 24} (2000), 623--644].
For the lower bound in the bipartite case, we had to extend the existence of the
midrange crossing constant from the class of all graphs to other graph classes, including the class of bipartite graphs. Likely there are infinitely many different
crossing constants for graph classes.
We gave a simple proof to the $ 8 \over 9\pi^2$ upper bound on the (ordinary) midrange crossing constant of
Pach and T\'oth [{\it Combinatorica} {\bf 17} (1997), 427--439] and Pach, Radoi\v ci\v c, Tardos and T\'oth [{\it DCG} {\bf 36} (2006), 527--552]
using spherical geometry.
This is joint work with John Asplund, \'Eva Czabarka, Gregory Clark,
Garner Cochran, Arran Hamm, Josiah Reiswig, Inne Singgih, Gwen Spencer, Libby Taylor and
Zhiyu Wang, originating from a Mathematics Research Communities program.