  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.
