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Ramsey Theory on the Integer Grid: The "L" Problem

Isaac Mammel, William Smith, Carl Yerger

TL;DR

This work investigates the least $n$ for which any $3$-coloring of the $n\times n$ grid contains a monochromatic $L$, a special $L$-shape defined by $\{(i, j), (i, j+t), (i+t, j+t)\}$. Building on a 2015 bound of $R_3(L) \le 2593$, the authors iteratively tighten the upper bound using diagonal interval analysis, non-consecutive red-point intervals, and Golomb-ruler arguments, culminating in a final bound of $R_3(L) \le 493$. They also examine the lower bound via SAT solvers, detailing several solving strategies (brute force, triangulation, column-fixings, and reverse-diagonal analyses) and reporting a satisfiable $20\times20$ grid with no monochromatic $L$. The results showcase a blend of combinatorial interval counting and computational search to advance Ramsey-type questions on grids, with open problems inviting further refinements and extensions to larger colorings. The methods have potential implications for understanding structured colorings on lattices and for developing SAT-based approaches to similar Ramsey-type grid problems.

Abstract

In an $[n] \times [n]$ integer grid, a monochromatic $L$ is any set of points $\{(i, j), (i, j+t), (i+t, j+t)\}$ for some positive integer $t$, where $1 \leq i, j, i+t, j+t \leq n$. In this paper, we investigate the upper bound for the smallest integer $n$ such that a $3$-colored $n \times n$ grid is guaranteed to contain a monochromatic $L$. We use various methods, such as counting intervals on the main diagonal and using Golomb rulers, to improve the upper bound. This bound originally sat at 2593, and we improve it first to 1803, then to 1573, then to 772, and finally to 493. In the latter part of this paper, we discuss the lower bound and our attempts to improve it using SAT solvers.

Ramsey Theory on the Integer Grid: The "L" Problem

TL;DR

This work investigates the least for which any -coloring of the grid contains a monochromatic , a special -shape defined by . Building on a 2015 bound of , the authors iteratively tighten the upper bound using diagonal interval analysis, non-consecutive red-point intervals, and Golomb-ruler arguments, culminating in a final bound of . They also examine the lower bound via SAT solvers, detailing several solving strategies (brute force, triangulation, column-fixings, and reverse-diagonal analyses) and reporting a satisfiable grid with no monochromatic . The results showcase a blend of combinatorial interval counting and computational search to advance Ramsey-type questions on grids, with open problems inviting further refinements and extensions to larger colorings. The methods have potential implications for understanding structured colorings on lattices and for developing SAT-based approaches to similar Ramsey-type grid problems.

Abstract

In an integer grid, a monochromatic is any set of points for some positive integer , where . In this paper, we investigate the upper bound for the smallest integer such that a -colored grid is guaranteed to contain a monochromatic . We use various methods, such as counting intervals on the main diagonal and using Golomb rulers, to improve the upper bound. This bound originally sat at 2593, and we improve it first to 1803, then to 1573, then to 772, and finally to 493. In the latter part of this paper, we discuss the lower bound and our attempts to improve it using SAT solvers.

Paper Structure

This paper contains 15 sections, 10 theorems, 22 equations, 6 figures, 3 algorithms.

Table of Contents

  1. Introduction to the Problem
  2. The Upper Bound
  3. Intervals
  4. Unaccounted Intervals
  5. Further Improvements
  6. The Lower Bound/On SAT Solvers
  7. Solving Methods
  8. Brute Force
  9. Triangulation
  10. The "Right Column" Method
  11. Fixing Values
  12. Reverse Diagonals
  13. Table of Solving Methods
  14. Open Problems
  15. Acknowledgements.

Key Result

Theorem 1

For all positive integers $k$, there exists a positive integer $n$ such that, for all $k$-colorings of $[n] \times [n]$, there is a monochromatic $L$. That is, there exist positive integers $x, y,$ and $d$ such that:

Figures (6)

  • Figure 1: An illustration of the contradiction we reach in Theorem $1$. Points with unspecified color are grey, and the point that cannot be any color without forming a monochromatic $L$ is black.
  • Figure 2: An illustration of how certain intervals force points on corresponding diagonals to be either blue or green. For example, two adjacent red points force a blue/green point on the subdiagonal right below.
  • Figure 3: A satisfiable $20 \times 20$ grid as found by CryptoMiniSAT. If the integer in the $i$th row and $j$th column is $c$, then the color of $(i, j)$ in our grid is color $c$. In this grid (as well as all subsequent grids in this paper), the $0$s are colored red, the $1$s are colored blue, and the $2$s are colored green.
  • Figure 4: A representation of an $n \times n$ grid, where each point $p$ is labeled with the number of $L$s that can be formed with $p$ as the uppermost point.
  • Figure 5: A picture of the main reverse diagonal (colored red) on a $5 \times 5$ integer grid.
  • ...and 1 more figures

Theorems & Definitions (20)

  • Theorem 1: witt
  • Theorem 2: canackisat
  • proof
  • Definition 1
  • Lemma 1
  • proof
  • Theorem 3
  • proof
  • Theorem 4
  • proof
  • ...and 10 more