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

TL;DR

The paper formulates Gallai-type questions for colorings of finite grids under homothety and general similarity, introducing $\Gamma(A,c)$ and $\Sigma(A,c)$ to denote minimal grid sizes forcing monochromatic copies of a finite $A\subset\mathbb{Z}^n$. It develops a SAT-based framework, $\Upsilon(S,\mathcal{A})$, to test the avoidance of monochromatic copies, and uses both managed brute force and advanced SAT/CDCL methods with symmetry breaking to compute successive bounds for triangles, squares, and rectangles in 2D, plus hexagons and cubes in higher settings. Key contributions include explicit Gallai numbers for triangles ($\Sigma(\triangle,2)=4$, $\Gamma(\triangle,2)=5$), squares ($\Sigma(\square,2)=7$, $\Gamma(\square,2)=15$), and several families of rectangles (e.g., $\Sigma(R_k,2)=8,13,15$ for $k=2,3,4$; $\Gamma(R_k,2)=27,40,52,66$ for $k=2,3,4,5$), as well as new lower bounds for hexagons and 3D cubes. The work further provides symmetry-class classifications of solutions and deduces density-type consequences for monochromatic figures in infinite lattices and the plane, illustrating the broad interplay between finite-combinatorics, SAT-solving, and discrete geometry.

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 $\Z^3$.

Figures (6)

• Figure 1: A 2-coloring of the $4 \times 4$ grid such that every horizontal square with vertices on the grid has exactly two white and two black vertices. This is a maximal solution for this problem.
• Figure 2: This colored grid $T_3$ does not contain any monochromatic horizontal, skew or reversed equilateral triangle. It is a maximal solution for this problem.
• Figure 3: This colored grid $T_4$ does not contain any monochromatic horizontal upwards oriented triangle. It is a maximal solution for this problem. We observe that the grid contains a skew equilateral triangle. Of course, as $\Sigma(\triangle, 2) = 4$, one couldn't avoid this.
• Figure 4: A 2-coloring of the $6 \times 6$ grid with no monochromatic horizontal or skew squares. This is a maximal solution for this problem.
• Figure 5: A 2-coloring of the $14 \times 14$ grid with no monochromatic horizontal squares. This is a maximal solution for this problem. But as a $14 \times 14$ grid can be divided in four disjoint $7 \times 7$ grids, and as $\Sigma(\square, 2)=7$, there are at least four skew monochromatic squares.

Theorems & Definitions (18)

• Theorem 1.1: Gallai
• Theorem 2.1
• Theorem 2.2
• Lemma 3.1
• Theorem 4.1
• Theorem 4.2
• Theorem 5.1
• Theorem 5.2: Walton - Li
• Theorem 6.1
• Theorem 6.2