Abstract
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.
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