Extension of colorings

Let K be a combinatorial (d−1)-sphere with vertices colored in n colors, n≥d+1. We prove that K bounds an n-colored combinatorial ball. This theorem generalizes previously known facts for d=2 and 3. A further generalization is obtained. Namely, let L be a simplicial complex of dimension d and K be a subcomplex of L. Then any vertex coloring of K in n≥d+1 colors extends to some subdivision of L relative to K. Besides, in both cases the extension can be required to use only d+1 of n colors in the complement to K.