Generic regularity in time for solutions of the Stefan problem in 4+1 dimensions

We show that the free boundary of a solution of the Stefan problem in ${\mathbb{R}}^{4+1}$ is a $3$-dimensional manifold of class $C^{\mathrm{\infty }}$ in ${\mathbb{R}}^4$ for almost every time. This is achieved by showing that for all dimensions $n$ the singular set $\mathrm{\Sigma }\subset {\mathbb{R}}^{n+1}$ can be decomposed in two parts $\mathrm{\Sigma }={\mathrm{\Sigma }}^{\mathrm{\infty }}\cup {\mathrm{\Sigma }}^{\ast }$, where ${\mathrm{\Sigma }}^{\mathrm{\infty }}$ is covered by one $(n-1)$-dimensional manifold of class $C^{\mathrm{\infty }}$ in ${\mathbb{R}}^{n+1}$ and its projection onto the time axis has Hausdorff dimension 0, while ${\mathrm{\Sigma }}^{\ast }$ is parabolically countably $(n-2)$-rectifiable.