Convex hull of face vectors of colored complexes

In this paper we verify a conjecture by Kozlov [D.N. Kozlov, Convex Hulls of f - and β -vectors, Discrete Comput. Geom. 18 (1997) 421–431], which describes the convex hull of the set of face vectors of r -colorable complexes on n vertices. As part of the proof we derive a generalization of Turán’s graph theorem.