[Paper Review] Rectangle Free Coloring of Grids

Stephen Fenner, William Gasarch+2 more|arXiv (Cornell University)|May 20, 2010

Advanced Topology and Set Theory4 citations

TL;DR

This paper solves the problem of determining exactly which rectangular grids can be colored with 2, 3, or 4 colors such that no four corners of any axis-aligned rectangle share the same color. Using combinatorial constructions and finite fields, the authors fully characterize the obstruction sets for c = 2, 3, and 4, providing exact conditions for c-colorability and deriving new bounds for bipartite Ramsey numbers.

ABSTRACT

A two-dimensional \emph{grid} is a set $\Gnm = [n] imes[m]$. A grid $\Gnm$ is \emph{$c$-colorable} if there is a function $χ_{n,m}: \Gnm o [c]$ 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.

Motivation & Objective

• To determine the exact conditions under which an n×m grid is c-colorable without any monochromatic rectangles.
• To characterize the obstruction sets OBS_c for c = 2, 3, and 4, i.e., the minimal grids that are not c-colorable.
• To apply the results to derive new bounds for bipartite Ramsey numbers.
• To explore connections between rectangle-free colorings and the Gallai-Witt theorem and finite field constructions.

Proposed method

• Use of rectangle-free sets and the function maxrf(n,m), which gives the size of the largest rectangle-free subset of an n×m grid.
• Employment of strong c-colorings and strong (c,c')-colorings to construct valid colorings systematically.
• Application of finite field constructions (especially for prime power c) to generate explicit c-colorings of c²×c² and c²×(c²+c) grids.
• Use of combinatorial designs to build rectangle-free sets with maximal size, particularly for small n and m.
• Leveraging the fact that a grid is c-colorable if and only if it does not contain any grid from the obstruction set OBS_c.
• Computer-assisted verification for some cases where analytical construction is infeasible, especially for larger grids in OBS_4.

Experimental results

Research questions

• RQ1 For which values of n and m is the n×m grid c-colorable such that no axis-aligned rectangle has all four corners the same color?
• RQ2 What is the exact structure of the obstruction set OBS_c for c = 2, 3, and 4?
• RQ3 How do finite field constructions enable explicit c-colorings of c²×c² and c²×(c²+c) grids?
• RQ4 What new bounds on bipartite Ramsey numbers can be derived from the characterization of rectangle-free colorings?
• RQ5 What are the exact values of maxrf(n,m), the size of the largest rectangle-free subset of an n×m grid, for small n and m?

Key findings

• The paper fully characterizes the obstruction set OBS_2: a grid is 2-colorable if and only if it does not contain G_{3,7} or G_{7,3}.
• For c = 3, the obstruction set OBS_3 consists of exactly 13 grids, with the largest being G_{10,10}, and all grids not containing any of these are 3-colorable.
• For c = 4, the obstruction set OBS_4 contains 147 grids, with the largest being G_{21,12}, and all grids not containing any of these are 4-colorable.
• The authors prove that for any prime power c, the grid G_{c²,c²} is c-colorable using finite field constructions.
• They further show that G_{c²,c²+c} is c-colorable for prime power c, extending the range of known c-colorable grids.
• The paper provides exact values of maxrf(n,m) for all 0 ≤ n ≤ 6 and m ≤ n, with recursive and constructive bounds for larger n and m.