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

We prove that when a compact mean-convex subset of R^n+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