Flag f-vectors of colored complexes

It is shown that conditions stronger in a certain sense than colorshifting cannot be placed on the class of colored complexes without changing the characterization of the flag f-vectors. In the late 1970s, Stanley [6] showed that two particular classes of simplicial complexes have equivalent characterizations of their flag f-vectors. Several years later, Björner, Frankl, and Stanley [1] showed that two additional classes of simplicial complexes shared this same characterization. Unfortunately, no one has a characterization for any of these classes of simplicial complexes, but we only know that characterizing one would suffice for all four. The two additional classes of simplicial complexes included in the equivalence of Björner, Frankl, and Stanley are each proper subsets of one of the classes of complexes in Stanley’s original paper. Thus, the paper of Björner, Frankl, and Stanley could be thought of as progress toward a solution by narrowing the class of complexes to consider. In this paper, we show in Theorem 5 that extending this approach to a solution of the problem by further narrowing one of the classes of complexes in a certain sense is impossible. Recall that a simplicial complex ∆ on a vertex set W is a collection of subsets of W such that (i) for every v ∈ W , {v} ∈ ∆ and (ii) for every B ∈ ∆, if A ⊂ B, then A ∈ ∆. The elements of ∆ are called faces. A face on i vertices is said to have dimension i− 1, while the dimension of a complex is the maximum dimension of a face of the complex. The i-th f-number of a simplicial complex ∆, fi−1(∆) is the number of faces of ∆ on i vertices. The f-vector of ∆ lists the f-numbers of ∆. One interesting question to ask is which integer vectors can arise as f-vectors of simplicial complexes. Much work has been done toward answering this for various classes of simplicial complexes. For example, the Kruskal-Katona theorem [5, 4] characterizes the fvectors of all simplicial complexes. In this paper, we wish to deal with colored complexes, where the coloring provides additional data. A coloring of a simplicial complex is a labeling of the vertices of the complex with colors such that no two vertices in the same face are the same color. Because any two vertices in a face are connected by an edge, this is equivalent to merely requiring that any two adjacent vertices be assigned different colors. If the set of colors has n colors, we refer to the colors as 1, 2, . . . , n. The set of colors is denoted by [n] = {1, 2, . . . , n}. The color set of a face is the subset of [n] consisting of the colors of the vertices of the face. The Frankl-Füredi-Kalai [2] theorem characterizes the f-vectors of all simplicial complexes that can be colored with n colors. We wish to use a refinement of the usual notion of f-vectors. The flag f-numbers of a colored simplicial complex ∆ on a color set [n] are defined by, for any subset