Embedding products of graphs into Euclidean spaces

Mikhail Skopenkov. "Embedding products of graphs into Euclidean spaces." Fundamenta Mathematicae 179.3 (2003): 191-198. <http://eudml.org/doc/283348>.

@article{MikhailSkopenkov2003,
abstract = {For any collection of graphs $G₁,...,G_N$ we find the minimal dimension d such that the product $G₁ × ... × G_N$ is embeddable into $ℝ^d$ (see Theorem 1 below). In particular, we prove that (K₅)ⁿ and $(K_\{3,3\})ⁿ$ are not embeddable into $ℝ^\{2n\}$, where K₅ and $K_\{3,3\}$ are the Kuratowski graphs. This is a solution of a problem of Menger from 1929. The idea of the proof is a reduction to a problem from so-called Ramsey link theory: we show that any embedding $Lk O → S^\{2n-1\}$, where O is a vertex of (K₅)ⁿ, has a pair of linked (n-1)-spheres.},
author = {Mikhail Skopenkov},
journal = {Fundamenta Mathematicae},
keywords = {Kuratowski criterion; van Kampen obstruction; Ramsey link theory},
language = {eng},
number = {3},
pages = {191-198},
title = {Embedding products of graphs into Euclidean spaces},
url = {http://eudml.org/doc/283348},
volume = {179},
year = {2003},
}

TY - JOUR
AU - Mikhail Skopenkov
TI - Embedding products of graphs into Euclidean spaces
JO - Fundamenta Mathematicae
PY - 2003
VL - 179
IS - 3
SP - 191
EP - 198
AB - For any collection of graphs $G₁,...,G_N$ we find the minimal dimension d such that the product $G₁ × ... × G_N$ is embeddable into $ℝ^d$ (see Theorem 1 below). In particular, we prove that (K₅)ⁿ and $(K_{3,3})ⁿ$ are not embeddable into $ℝ^{2n}$, where K₅ and $K_{3,3}$ are the Kuratowski graphs. This is a solution of a problem of Menger from 1929. The idea of the proof is a reduction to a problem from so-called Ramsey link theory: we show that any embedding $Lk O → S^{2n-1}$, where O is a vertex of (K₅)ⁿ, has a pair of linked (n-1)-spheres.
LA - eng
KW - Kuratowski criterion; van Kampen obstruction; Ramsey link theory
UR - http://eudml.org/doc/283348
ER -