Coloring grids

Ramiro de la Vega. "Coloring grids." Fundamenta Mathematicae 228.3 (2015): 283-289. <http://eudml.org/doc/286401>.

@article{RamirodelaVega2015,
abstract = {A structure $ = (A;E_\{i\})_\{i∈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 χ: A → n is an acceptable coloring if for all i ∈ n, the $χ^\{-1\}(i)$ intersects each $E_\{i\}$-equivalence class in a finite set. If B is a set, then the n-cube Bⁿ may be seen as an n-grid, where the equivalence classes of $E_\{i\}$ are the lines parallel to the ith 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 does not admit an acceptable coloring, then every finite n-cube is embeddable in .},
author = {Ramiro de la Vega},
journal = {Fundamenta Mathematicae},
keywords = {continuum hypothesis; Sierpiński’s theorem; -grids},
language = {eng},
number = {3},
pages = {283-289},
title = {Coloring grids},
url = {http://eudml.org/doc/286401},
volume = {228},
year = {2015},
}

TY - JOUR
AU - Ramiro de la Vega
TI - Coloring grids
JO - Fundamenta Mathematicae
PY - 2015
VL - 228
IS - 3
SP - 283
EP - 289
AB - A structure $ = (A;E_{i})_{i∈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 χ: A → n is an acceptable coloring if for all i ∈ n, the $χ^{-1}(i)$ intersects each $E_{i}$-equivalence class in a finite set. If B is a set, then the n-cube Bⁿ may be seen as an n-grid, where the equivalence classes of $E_{i}$ are the lines parallel to the ith 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 does not admit an acceptable coloring, then every finite n-cube is embeddable in .
LA - eng
KW - continuum hypothesis; Sierpiński’s theorem; -grids
UR - http://eudml.org/doc/286401
ER -