Vertex colorings of simplicial complexes

We define an (r,s)-coloring of an abstract simplicial complex to be a coloring using r colors of the vertices so that in any simplex at most s vertices have the same color. We translate the problem of finding an (r,s)-coloring of a given simplex to a topological problem and to a ring theoretic problem. The topological version involves vector bundles over Davis–Januszkiewicz spaces and the algebraic one involves factorizations in Stanley-Reisner rings.