Abstract: 
For any collection of graphs $G_1,\dots,G_N$ we find the minimal dimension $d$ such that the product $G_1\times \dots\times G_N$ is embeddable into ${{\fam \msyfam \relax R}}^d$ (see Theorem\penalty \@M \ 1 below). In particular, we prove that $(K_5)^n$ and $(K_{3,3})^n$ are not embeddable into ${{\fam \msyfam \relax R}}^{2n}$, where $K_5$ 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 $\mathop {\fam \z@ \tenrm Lk}\nolimits O\to S^{2n-1}$, where $O$ is a vertex of $(K_5)^n$, has a pair of linked $(n-1)$-spheres.

1. Department of Differential Geometry
   Faculty of Mechanics and Mathematics
   Moscow State University
   Moscow, 119992, Russia