Embedding products of graphs into Euclidean spaces
Abstract
For any collection of graphs we find the minimal dimension d such that the product of these graphs is embeddable into the d-dimensional Euclidean space. In particular, we prove that the n-th powers of the Kuratowsky graphs are not embeddable into the 2n-dimensional Euclidean space. This is a solution of a problem of Menger from 1929. The idea of the proof is the reduction to a problem from so-called Ramsey link theory: we show that any embedding of L into the (2n-1)-dimensional sphere, where L is the join of n copies of a 4-point set, has a pair of linked (n-1)-dimensional spheres.
Cite
@article{arxiv.0808.1199,
title = {Embedding products of graphs into Euclidean spaces},
author = {Mikhail Skopenkov},
journal= {arXiv preprint arXiv:0808.1199},
year = {2025}
}
Comments
in English and Russian, 8 pages, 2 figures. Some standard material has been included when no suitable reference has been found (a definition of the general position, a proof of the parity lemma, a formula for the change of intersection index upon homotopy, a relation of intersection index to the linking number). A reference to a recent popular-science exposition has been added