Consider colorings of n-dimensional complexes. Given a complex $C$, and a coloring of $C$, a simplex $\alpha \in C$ is said (n-x)-complete with respect to this coloring if all its vertices receive at least $(n-x)$ colors.
Sperner's Lemma and the Index lemma relate the number of n-complete simplices in a complex with the number of n-complete simplices in its boundary.
I need to generalize this result to relate the number of $(n-x)$-complete simplices in a complex and their number in its boundary.
Does this generalization already exists ? Any help is welcome.
Thank you.