Pick $n\ge 3$ non-vertical lines $\mathscr{L}:=\{\ell_1,\ldots,\ell_n\}$ in the plane which are pairwise non-parallel, and they are not all concurrent in a single point. 

>**Question.** Does there exist a vertical line $L$ which has $n-1$ or $n-2$ intersections with $\mathscr{L}$?

*Note:* More explicitly, "$L$ has $k$ intersections with $\mathscr{L}$" means that the set $\{L\cap \ell_i: i=1,\ldots,n\}$ has $k$ distinct points.

*Partial note 1:* By using a dual result due to Ungar [here][1] one can see that there are at least $n-1$ vertical lines $L$ which have at most $n-1$ intersections with $\mathscr{L}$

*Partial note 2:* The [Silvester-Gallai problem][2] shows the existence of an ordinary line, that is, given $n$ points in the plane, the exists a connecting line passing through exactly two of those points (this has been improved [here][3] by Green and Tao, solving a conjecture of Erdos). In the setting above, dually, there exists a point which is the intersection of exactly two lines of $\mathscr{L}$. 

*Partial note 3:* This seems to be somehow related to (dual of) Problem 1 at the end of [this article][4], as suggested by domotorp in his answer at [this previous][5] question. However, the motivation of the latter problem was a conjecture of Erdos and Purdy, namely,  $n$ points in the plane determine at least $\lfloor (n-1)/2\rfloor$ triangles with distinct positive area; finally, this has been proved by Pinchasi in 2008 [here][6]. 


  [1]: https://www.sciencedirect.com/science/article/pii/0097316582900450
  [2]: https://en.wikipedia.org/wiki/Sylvester%E2%80%93Gallai_theorem
  [3]: https://link.springer.com/article/10.1007/s00454-013-9518-9
  [4]: http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.86.9181&rep=rep1&type=pdf
  [5]: https://mathoverflow.net/questions/412876/parallel-lines-containing-a-subset-with-even-cardinality
  [6]: https://epubs.siam.org/doi/10.1137/07068299X