Coloring Grids

TL;DR

This paper characterizes when certain grid-like structures with multiple equivalence relations admit special colorings, and shows that non-colorable grids contain all finite cubes as substructures.

Contribution

It provides a characterization of $n$-grids with acceptable colorings using elementary submodels and links non-colorability to the embeddability of all finite $n$-cubes.

Findings

Elementary submodels characterize $n$-grids with acceptable colorings.

Non-colorable $n$-grids embed all finite $n$-cubes.

The approach connects coloring properties with structural embeddability.

Abstract

A structure $\mathcal{A}=\left(A;E_i\right)_{i\in n}$ where each $E_i$ is an equivalence relation on $A$ is called an $n$-grid if any two equivalence classes coming from distinct $E_i$'s intersect in a finite set. A function $\chi: A \to n$ is an acceptable coloring if for all $i \in n$, the set $\chi^{-1}(i)$ intersects each $E_i$-equivalence class in a finite set. If $B$ is a set, then the $n$-cube $B^n$ may be seen as an $n$-grid, where the equivalence classes of $E_i$ are the lines parallel to the $i$-th coordinate axis. We use elementary submodels of the universe to characterize those $n$-grids which admit an acceptable coloring. As an application we show that if an $n$-grid $\mathcal{A}$ does not admit an acceptable coloring, then every finite $n$-cube is embeddable in $\mathcal{A}$.