Rectangle Free Coloring of Grids

TL;DR

This paper characterizes exactly which two-dimensional grids can be colored with 2, 3, or 4 colors without forming a monochromatic rectangle, advancing understanding in combinatorics and Ramsey theory.

Contribution

It provides exact criteria for grid colorability with 2, 3, and 4 colors, using combinatorics, finite fields, and tournament graphs.

Findings

Precisely identified 2-colorable grids.

Precisely identified 3-colorable grids.

Precisely identified 4-colorable grids.

Abstract

A two-dimensional \emph{grid} is a set $\Gnm = [n]\times[m]$. A grid $\Gnm$ is \emph{$c$-colorable} if there is a function $\chi_{n,m}: \Gnm \to [c]$ such that there are no rectangles with all four corners the same color. We address the following question: for which values of $n$ and $m$ is $\Gnm$ $c$-colorable? This problem can be viewed as a bipartite Ramsey problem and is related to a the Gallai-Witt theorem (also called the multidimensioanl Van Der Waerden's Theorem). We determine (1) \emph{exactly} which grids are 2-colorable, (2) \emph{exactly} which grids are 3-colorable, and (3) \emph{exactly} which grids are 4-colorable. We use combinatorics, finite fields, and tournament graphs.