Define the chromatic number $\chi(H)$ of a (hyper)graph $H$ as the smallest $k$ such that its vertices can be $k$-colored such that no (hyper)edge is monochromatic.  
Given a collection of points and lines, $P$ and $L$, consider the hypergraph $H(P,L)$ on the vertex set $P$ such that the edges are formed by the collinear points.  
Define the graph $G(P,L)$ as the restriction of $H(P,L)$ to its edges of size 2.  

>By what function of $\chi(G)$ can we bound $\chi(H)$ from above?

Note: The [dual of this question][1] was asked by a [new and anonymous user][2] 2 days ago, who shortly afterwards deleted it.


  [1]: https://mathoverflow.net/questions/439177/chromatic-number-of-a-dual-graph-of-straight-lines
  [2]: https://mathoverflow.net/users/496369/causalityrefilm