The colored Kruskal-Katona theorem, which extends the Kruskal-Katona theorem, is equivalent to a numerical characterization of the f-vectors of colored simplicial complexes. The underlying theme is the study of initial sets of the reverse lexicographical order. In this talk, I will give a generalization of the colored Kruskal-Katona theorem, explain its relation to the study of initial sets, and discuss its consequences for the f-vectors of colored complexes and balanced complexes of arbitrary type.