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.

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 $d$.

Full text

See pages 1-last of FinalPaper.pdf