Characterizing f-vectors (Supplemental)

Abstract

What characterizes an f-vector of a simplicial complex? The entries are obviously nonnegative integers, and f ₀ = 1, but what other restrictions are there? Well, for one thing, if there are n vertices there can be at most $\binom{n}{2}$ edges, since there is at most one edge for every pair of vertices. That is,

$$\displaystyle{f_{2} \leq \binom{f_{1}}{2}.}$$

This simple observation can be greatly generalized. It turns out there is a sharp upper bound on the number of (k + 1)-faces expressed as a polynomial in f _k . (Likewise, there is a sharp lower bound on the number of k faces required for a given number of (k + 1)-faces.) Collectively, these restrictions, known as the Kruskal-Katona-Schützenberger inequalities (or KKS inequalities), characterize the set of f-vectors of simplicial complexes. We remark that characterizing f-vectors of boolean complexes is much, much simpler. See Problem 8.7 (Fig. 10.1).