Erd\H{o}s - Szekeres Theorem for Lines

TL;DR

This paper extends the Erd ext{"o}s-Szekeres theorem from points to lines, establishing bounds for the minimum number of lines in general position needed to guarantee a subset in convex position, and explores related variants.

Contribution

It provides the first bounds for the line version of the Erd ext{"o}s-Szekeres theorem, including dual variants and generalizations, with nearly matching upper and lower bounds.

Findings

Established bounds for line sets in convex position

Proved dual versions of the theorem and its variants

Compared bounds with the point case, highlighting differences

Abstract

According to the Erd\H{o}s-Szekeres theorem, for every $n$, a sufficiently large set of points in general position in the plane contains $n$ in convex position. In this note we investigate the line version of this result, that is, we want to find $n$ lines in convex position in a sufficiently large set of lines that are in general position. We prove almost matching upper and lower bounds for the minimum size of the set of lines in general position that always contains $n$ in convex position. This is quite unexpected, since in the case of points, the best known bounds are very far from each other. We also establish the dual versions of many variants and generalizations of the Erd\H os-Szekeres theorem.