Avoiding Monochromatic Rectangles Using Shift Patterns

Abstract

Ramsey Theory deals with avoiding certain patterns. When constructing an instance that avoids one pattern, it is observed that other patterns emerge. For example, repetition emerges when avoiding arithmetic progression (Van der Waerden numbers), while reflection emerges when avoiding monochromatic solutions of $a+b=c$ (Schur numbers). We exploit observed patterns when coloring a grid while avoiding monochromatic rectangles. Like many problems in Ramsey Theory, this problem has a rapidly growing search space that makes computer search difficult. Steinbach et al. obtained a solution of an 18 by 18 grid with 4 colors by enforcing a rotation symmetry. However, that symmetry is not suitable for 5 colors. In this article, we will encode this problem into propositional logic and enforce so-called internal symmetries, which preserves satisfiability, to guide SAT-solving. We first observe patterns with 2 and 3 colors, among which the "shift pattern" can be easily generalized and efficiently encoded. Using this pattern, we obtain a new solution of the 18 by 18 grid that is non-isomorphic to the known solution. We further analyze the pattern and obtain necessary conditions to further trim down the search space. We conclude with our attempts on finding a 5-coloring of a 26 by 26 grid, as well as further open problems on the shift pattern.