Abstract
We present examples of flag homology spheres whose γ-vectors satisfy the Kruskal–Katona inequalities. This includes several families of well-studied simplicial complexes, including Coxeter complexes and the simplicial complexes dual to the associahedron and to the cyclohedron. In these cases, we construct explicit flag simplicial complexes whose f-vectors are the γ-vectors in question, and so a result of Frohmader shows that the γ-vectors satisfy not only the Kruskal–Katona inequalities but also the stronger Frankl–Füredi–Kalai inequalities. In another direction, we show that if a flag (d−1)-sphere has at most 2d+3 vertices its γ-vector satisfies the Frankl–Füredi–Kalai inequalities. We conjecture that if Δ is a flag homology sphere then γ(Δ) satisfies the Kruskal–Katona, and further, the Frankl–Füredi–Kalai inequalities. This conjecture is a significant refinement of Gal’s conjecture, which asserts that such γ-vectors are nonnegative.
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Research of the first author partially supported by an NSF Award DMS-0757828.
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Nevo, E., Petersen, T.K. On γ-Vectors Satisfying the Kruskal–Katona Inequalities. Discrete Comput Geom 45, 503–521 (2011). https://doi.org/10.1007/s00454-010-9243-6
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DOI: https://doi.org/10.1007/s00454-010-9243-6