Intersections of vertical lines with non-vertical lines

TL;DR

The paper investigates the intersection properties of vertical lines with families of non-vertical, non-parallel lines in the plane, providing a negative answer to a specific geometric question about the number of intersection points.

Contribution

It demonstrates that for certain configurations of lines, no vertical line intersects the union in exactly n-1 or n-2 points, answering a previously open question.

Findings

No vertical line intersects the union in exactly n-1 points.

No vertical line intersects the union in exactly n-2 points.

Counterexamples to the posed question are provided.

Abstract

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