On γ-vectors satisfying the Kruskal–Katona inequalities, Discrete Comput

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 com-plexes and the simplicial complexes dual to the associahedron and to the cyclohedron. In another direction, we show that if a flag (d − 1)-sphere has at most 2d+ 2 vertices its γ-vector satisfies the Kruskal-Katona in-equalities. We conjecture that if ∆ is a flag homology sphere then γ(∆) satisfies the Kruskal-Katona inequalities. This conjecture is a signifi-cant refinement of Gal’s conjecture, which asserts that such γ-vectors are nonnegative. 1