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Frankl-Füredi-Kalai inequalities on the $$\gamma$$-vectors of flag nestohedra. (English) Zbl 1307.52006
A building set $$\mathcal{B}$$ on $$[n] = \{1,\ldots,n\}$$ is a set of nonempty subsets of $$[n]$$ such that (i) for any non-disjoint $$I,J\in \mathcal{B}$$ their union $$I\cup J$$ is in $$\mathcal{B}$$, and (ii) $$\mathcal{B}$$ contains all singleton sets $$\{i\}$$ for each $$i\in [n]$$. The corresponding nestohedron $$P_{\mathcal{B}}$$ is a polytope defined as the Minkowski sum $$P_{\mathcal{B}} := \sum_{I\in \mathcal{B}}\Delta_I$$, where $$\Delta_I$$ is the standard simplex with vertex set $$\{\tilde{e}_i : i\in I\}$$ where $$\tilde{e}_i$$ denotes the standard $$i$$-th basis vector for $${\mathbb{R}}^n$$. A polytope is flag if every collection of pairwise intersecting facets has a nonempty intersection (i.e. the collection of the facets of $$P$$ has the Helly property). A simplicial complex is flag if every set of pairwise adjacent vertices is a face.
The main result in this paper states that if $$P_{\mathcal{B}}$$ is a flag nestohedron, then there is a corresponding flag simplicial complex $$\Gamma(\mathcal{B})$$ whose $$f$$-vector $$f(\Gamma(\mathcal{B}))$$ equals that of the $$\gamma$$-vector $$\gamma(P_{\mathcal{B}})$$ of $$P_{\mathcal{B}}$$. This shows, in particular, that $$\gamma(P_{\mathcal{B}})$$ satisfies the Frankl-Füredi-Kalai inequalities [P. Frankl et al., Math. Scand. 63, No. 2, 169–178 (1988; Zbl 0651.05003)].

##### MSC:
 52B05 Combinatorial properties of polytopes and polyhedra (number of faces, shortest paths, etc.)
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