Linear coloring conjecture for touching sets of strings

A touching set of strings is a finite family of strings in the plane in which no pair crosses, and it is $k$-touching if every point of the plane belongs to at most $k$ strings. Linear coloring conjecture. There is a constant $c>0$ such that every $k$-touching set of strings can be colored with $ck$ colors. The conjecture was subsequently proved by Fox and Pach, who obtained the bound $6ek+1$ colors.