We prove the following conjecture of K.T. Atanassov Let $T$ be a triangulation of a $d$-dimensional polytope $P$ with $n$ vertices $\V_1,\V_2,\dots,\V_n$. Label the vertices of $T$ by $1,2,\dots, n$ in such a way that a vertex of $T$ belonging to the interior of a face $F$ of $P$ can only be labeled by $j$ if $\V_j$ is on $F$. Then there are at least $n-d$ full dimensional simplices of $T$, each labeled with $d+1$ different labels. We provide two proofs of this result: a non-constructive proof introducing the notion of a {\em pebble set} of a polytope, and a constructive proof using a path-following argument. Our non-constructive proof has interesting relations to minimal simplicial covers of convex polyhedra and their chamber complexes.