Two-colorings of finite grids: variations on a theorem of Tibor Gallai

TL;DR

This paper investigates the minimal grid size needed to guarantee monochromatic homothetic or similar images of certain shapes in finite colorings of integer grids, extending Gallai's theorem with computational methods.

Contribution

It extends Gallai's theorem to similarity transformations, computes specific Gallai similarity numbers, and employs SAT solvers for computational proofs.

Findings

Gallai similarity numbers for lattice rectangles of size 1×k for k=2,3,4

New lower bounds for regular hexagons on triangular lattice

Lower bounds for three-dimensional cubes in Z^3

Abstract

A celebrated but non-effective theorem of Tibor Gallai states that for any finite set $A$ of $\Z^n$ and for any finite number of colors $c$ there is a minimal $m$ such that no coloring of the finite $m^n$-grid can avoid that a homothetic image of $A$ is monochromatic. We find (or confirm) $m$ for equilateral triangles, squares, and various types of rectangles. Also, we extend the problem from homothety to general similarity, or to similarity generated using some special rotations. In particular, we compute Gallai similarity numbers for lattice rectangles similar to $1\times k$ (in all orientations) for $k=2,3,4$. The solutions have been found in the framework of the Satisfiability Problem in Propositional Logic (SAT). While some questions were solved using managed brute force, for the more computationally intensive questions we used modern SAT solvers together with symmetry breaking techniques. Some other minor questions are solved for triangles and squares, and new lower bounds are found for regular hexagons on the triangular lattice and for three-dimensional cubes in $\Z^3$.