Happy Ending or Many Concurrent Lines

TL;DR

This paper extends the Erdős-Szekeres convex polygon problem to lines in the plane, establishing bounds for the minimum number of lines needed to guarantee either a set of concurrent lines or a convex position.

Contribution

It provides new upper and lower bounds for the line version of the Erdős-Szekeres problem, advancing understanding of geometric configurations of lines.

Findings

Established bounds for $ES_L(l,n)$

Extended Erdős-Szekeres problem to lines

Contributed to geometric combinatorics

Abstract

For each $n \geq 2$, $l \geq 3$, let ${ES}_L (l,n)$ be the minimum $N$ such that every family of $N$-lines in the plane contains either $l$ concurrent lines or $n$ lines in convex position. In this papar, we give the upper and lower bounds for $ES_L (l,n)$. This is one of the extensions of the line version of Erd\"{o}s-Szekeres convex polygon problem.