The linear edge bound conjecture for k-quasiplanar graphs

A $k$-quasiplanar graph is a graph that can be drawn in the plane with no $k$ pairwise crossing edges. The linear edge bound conjecture. For any $k>1$, a $k$-quasiplanar graph on $n$ vertices can have at most

$$c_k n$$

edges for some constant $c_k>0$. This is a long-standing conjecture about the extremal density of quasiplanar graphs; the paper does not state a resolution, so the conjecture remains open.