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

TL;DR

This paper studies the minimal grid size needed to guarantee a monochromatic L-shape in a 3-colored grid, improving upper bounds significantly through combinatorial methods and exploring lower bounds with SAT solvers.

Contribution

It introduces new upper bounds for the smallest grid size guaranteeing a monochromatic L, using innovative counting and Golomb ruler techniques, and discusses lower bound improvements via SAT solvers.

Findings

Upper bound improved from 2593 to 493

Used counting intervals and Golomb rulers for bounds

Explored lower bounds with SAT solvers

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.