Suppose $G$ is a graph with $m=|E(G)|$ edges and $n=|V(G)|$ vertices.
Suppose $sim(G)$, the simplification of $G$ contains $ m'  >> 3n $ edges.  
Call the set of edges corresponding to an edge $uv$ of $G$ the parallel set $p(uv)$ of $uv$. 
What is the maximum $r$ such that for some set $  W $ of edges of $sim(G)$  the set $U$
of edges $e$ such that each edge of $p(e)$ is intersected by some edge of $p(f)$ for some $f \in W$ has size at least $r|W|$?

What is the maximum $r$ such that for some set $  W $ of vertices of $sim(G)$  the set $U$
of edges $e \in sim(G)$ such that each edge of $p(e)$ is intersected by some edge of $p(f)$ for some edge $f$ with at least one endpoint in $W$ has size at least $r|W|$?

I found some stuff here but it has requirements on embedding https://arxiv.org/pdf/1801.00721.pdf