Sperner labellings: A combinatorial approach

In 2002, De Loera, Peterson and Su proved the following conjecture of Atanassov: let T be a triangulation of a d-dimensional polytope P with n vertices v₁, v₂, ..., v_n; label the vertices of T by 1, 2, ..., n in such a way that a vertex of T belonging to the interior of a face F of P can only be labelled by j if v_j is on F; then there are at least n - d simplices labelled with d + 1 different labels. We prove a generalisation of this theorem which refines this lower bound and which is valid for a larger class of objects.