Balanced partitions of 3-colored geometric sets in the plane

TL;DR

This paper investigates balanced partitions of 3-colored geometric sets in the plane, establishing existence results for segments, intervals, and rays that partition points and lines into balanced subsets.

Contribution

It introduces new geometric partitioning theorems for 3-colored sets, including segments intersecting lines of each color, intervals on curves, and rays dividing lattice points into balanced regions.

Findings

Existence of segments intersecting exactly one line of each color in 3-colored line arrangements.

Existence of disjoint intervals on a Jordan curve containing exactly k points of each color.

Existence of two rays with a common apex splitting lattice points into balanced subsets.

Abstract

Let $S$ be a finite set of geometric objects partitioned into classes or \emph{colors}. A subset $S'\subseteq S$ is said to be \emph{balanced} if $S'$ contains the same amount of elements of $S$ from each of the colors. We study several problems on partitioning $3$-colored sets of points and lines in the plane into two balanced subsets: (a) We prove that for every 3-colored arrangement of lines there exists a segment that intersects exactly one line of each color, and that when there are $2m$ lines of each color, there is a segment intercepting $m$ lines of each color. (b) Given $n$ red points, $n$ blue points and $n$ green points on any closed Jordan curve $\gamma$, we show that for every integer $k$ with $0 \leq k \leq n$ there is a pair of disjoint intervals on $\gamma$ whose union contains exactly $k$ points of each color. (c) Given a set $S$ of $n$ red points, $n$ blue points and $n$ green points in the integer lattice satisfying certain constraints, there exist two rays with common apex, one vertical and one horizontal, whose union splits the plane into two regions, each one containing a balanced subset of $S$.