For smooth knottings of compact (not necessarily orientable) n-dimensional manifolds in ${ℝ}^{n+2}$ (or ${}^{n+2}$), we generalize the notion of knot moves to higher dimensions. This reproves and generalizes the Reidemeister moves of classical knot theory. We show that for any dimension there is a finite set of elementary isotopies, called moves, so that any isotopy is equivalent to a finite sequence of these moves.

For any collection of graphs $G₁,...,G_N$ we find the minimal dimension d such that the product $G₁\times ...\times G_N$ is embeddable into ${ℝ}^d$ (see Theorem 1 below). In particular, we prove that (K₅)ⁿ and $\left(K_{3,3}\right)ⁿ$ 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 $LkO\to S^{2n-1}$, where O is a vertex of (K₅)ⁿ, has a pair of linked (n-1)-spheres.