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 ddimensional Euclidean space. In particular, we prove that the nth powers of the Kuratowsky graphs are not embeddable into the 2ndimensional 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 socalled Ramsey link theory: we show that any embedding of L into the (2n1)dimensional sphere, where L is the join of n copies of a 4point set, has a pair of linked (n1)dimensional spheres.
 Publication:

arXiv eprints
 Pub Date:
 August 2008
 arXiv:
 arXiv:0808.1199
 Bibcode:
 2008arXiv0808.1199S
 Keywords:

 Mathematics  Geometric Topology;
 57Q35;
 57Q45
 EPrint:
 in English and in Russian, 5 pages, 2 figures. Minor improvement of exposition, a reference to a popularscience introduction added