I've been trying to construct a proof for this exercise but I don't know where to start:
Find an infinite family of connected graphs and universal constants $c$ > 0 and $\epsilon$ > 0 such that the number of edges in each graph is at most $cn$, where $n$ is the number of vertices, but any drawing in the plane has at least $\epsilon n^{2}$ crossings.
I've tried different approaches such as doing induction on edges and crossing numbers, but none of them seem convenient. I'd appreciate any hints/ suggestions for the solution.
There is a Kleitman formula $$ cr(K_{5,s})=4\left[\frac{s}{2}\right]\left[\frac{s-1}{2}\right] $$ Since $|E(K_{5,s})|=5s$ and $n=|V(K_{5,s})|=s+5$ it follows from the Kleitman formula that we can take $c=5$ and $\epsilon=1/4$. Then the inequality $cr(K_{5,s})\geq (s+5)^2/4=n^2/4$ holds for $s\geq7$.