# On Partitions of Two-Dimensional Discrete Boxes

## TL;DR

This paper establishes a near-optimal lower bound on the number of sub-boxes needed in a partition of a 2D discrete box to satisfy a line-intersection property, with applications to graph colorings.

## Contribution

It introduces a new combinatorial bound for partitions with the $(k, \, \ell)$-piercing property, connecting geometric partitions to graph coloring problems.

## Key findings

- Derived a lower bound of $(k-1)+(\ell-1)+\lceil 2\sqrt{(k-1)(\ell-1)} \rceil$ for the number of sub-boxes.
- Showed the bound is nearly sharp for all positive integers $k, \ell$.
- Connected the geometric partition problem to a graph coloring problem with clique constraints.

## Abstract

Let $A$ and $B$ be finite sets and consider a partition of the \emph{discrete box} $A \times B$ into \emph{sub-boxes} of the form $A' \times B'$ where $A' \subset A$ and $B' \subset B$. We say that such a partition has the $(k,\ell)$-piercing property for positive integers $k$ and $\ell$ if every \emph{line} of the form $\{a\} \times B$ intersects at least $k$ sub-boxes and every line of the form $A \times \{b\}$ intersects at least $\ell$ sub-boxes. We show that a partition of $A \times B$ that has the $(k, \ell)$-piercing property must consist of at least $(k-1)+(\ell-1)+\left\lceil 2\sqrt{(k-1)(\ell-1)} \right\rceil$ sub-boxes. This bound is nearly sharp (up to one additive unit) for every $k$ and $\ell$. As a corollary we get that the same bound holds for the minimum number of vertices of a graph whose edges can be colored red and blue such that every vertex is part of red $k$-clique and a blue $\ell$-clique.