Intersections of vertical lines with non-vertical lines

Abstract

We answer negatively a question: if $\mathcal{F}$ is a family of $n⩾3$ non-vertical, pairwise non-parallel lines on the plane and $\bigcap \mathcal{F}=\varnothing$, is there a vertical line $L$ such that $L\cap \bigcup \mathcal{F}$ has exactly $n-1$ or $n-2$ points?

arXiv commentI solved the problem while I was participating at the conference in Będlewo. The problem is at the elementary level and can be applied to teach duality of points and lines