# On Equal Point Separation by Planar Cell Decompositions

## TL;DR

This paper explores how to separate points in the plane evenly using lines, revealing obstacles posed by certain Jordan curves and extending bounds related to the Erdős-Szekeres Conjecture.

## Contribution

It introduces new bounds on point separation and characterizes obstacles involving Jordan curves with low stabbing number, generalizing recent combinatorial results.

## Key findings

- Large point sets contain subsets on Jordan curves with bounded stabbing number
- Obstacles to equal separation are characterized by low stabbing number curves
- Results extend bounds related to the Erdős-Szekeres Conjecture

## Abstract

In this paper, we investigate the problem of separating a set $X$ of points in $\mathbb{R}^{2}$ with an arrangement of $K$ lines such that each cell contains an asymptotically equal number of points (up to a constant ratio). We consider a property of curves called the stabbing number, defined to be the maximum countable number of intersections possible between the curve and a line in the plane. We show that large subsets of $X$ lying on Jordan curves of low stabbing number are an obstacle to equal separation. We further discuss Jordan curves of minimal stabbing number containing $X$. Our results generalize recent bounds on the Erd\H{o}s-Szekeres Conjecture, showing that for fixed $d$ and sufficiently large $n$, if $|X| \ge 2^{c_dn/d + o(n)}$ with $c_d = 1 + O(\frac{1}{\sqrt{d}})$, then there exists a subset of $n$ points lying on a Jordan curve with stabbing number at most $d$.