# Planar Point Sets Determine Many Pairwise Crossing Segments

## TL;DR

This paper proves that any set of n points in general position in the plane determines nearly n^{1} pairwise crossing segments, significantly improving previous bounds, with a constructive proof applicable to dense geometric graphs.

## Contribution

It establishes a nearly linear lower bound on the number of pairwise crossing segments determined by planar point sets, extending to dense geometric graphs, and provides a constructive proof.

## Key findings

- Any n-point set in general position determines n^{1-o(1)} crossing segments.
- The result improves the previous lower bound of (\u221a n).
- The proof is fully constructive and applies to dense geometric graphs.

## Abstract

We show that any set of $n$ points in general position in the plane determines $n^{1-o(1)}$ pairwise crossing segments. The best previously known lower bound, $\Omega\left(\sqrt n\right)$, was proved more than 25 years ago by Aronov, Erd\H os, Goddard, Kleitman, Klugerman, Pach, and Schulman. Our proof is fully constructive, and extends to dense geometric graphs.