The size of the singular set in mean curvature flow of mean-convex sets

We prove that when a compact mean-convex subset of Rⁿ⁺¹ (or of an $(n+1)$-dimensional riemannian manifold) moves by mean-curvature, the spacetime singular set has parabolic hausdorff dimension at most $n-1$. Examples show that this is optimal. We also show that, as t→ ∞, the surface converges to a compact stable minimal hypersurface whose singular set has dimension at most $n - 7$. If $n < 7$, the convergence is everywhere smooth and hence after some time T, the moving surface has no singularities