Multicover inequalities on colored complexes

Suppose that C is a balanced simplicial complex. We show that its flag f-vector contains an interesting multiplicative structure. We define η _s (C):= log₂ f _S (C), and characterize the convex cone in which this flag η-vector may lie. Additionally, we specialize our results to the case when C is a pure balanced simplicial complex, and when C is a graded poset.