TITLE: Colorings of triangulations as a generalization of the four-color problem
QUESTION [8 upvotes]: My question can be viewed as a generalization of the four-color problem. Instead of a planar graph, consider a triangulation of a d-sphere. One wants to color vertices with N colors so that no two vertices connected by a 1-simplex have the same color. For a fixed d, is there an N which suffices for any triangulation? What are known upper bounds on such N?

REPLY [3 votes]: Already for $d=3$ the boundary complex of the cyclic polytope with n vertices has a complete graph and hence requires $n$ colors. 
You may ask if restricting the class of triangulations can lead to interesting extension of the FCT. Some answers in Generalizations of the Four-Color theorem are relevant.