On topological graphs with at most four crossings per edge

We show that if a graph $G$ with $n\ge 3$ vertices can be drawn in the plane such that each of its edges is involved in at most four crossings, then $G$ has at most $6n-12$ edges. This settles a conjecture of Pach, Radoi\v{c}i'{c}, Tardos, and T'oth, and yields a better bound for the famous Crossing Lemma: The crossing number, $cr(G)$, of a (not too sparse) graph $G$ with $n$ vertices and $m$ edges is at least $c\frac{m^3}{n^2}$, where $c>1\mathrm{/}29$. This bound is known to be tight, apart from the constant $c$ for which the previous best lower bound was $1\mathrm{/}31.1$. As another corollary we obtain some progress on the Albertson conjecture: Albertson conjectured that if the chromatic number of a graph $G$ is $r$, then $\mbox{cr}(G) \geq \mbox{cr}(K_r)$\mathrm{.}ThiswasverifiedbyAlbertson,Cranston,andFoxfor$r \leq 12$,andfor$r \leq 16$ by Bar'at and T'oth. Our results imply that Albertson conjecture holds for $r\le 18$.