On $γ$-vectors satisfying the Kruskal-Katona inequalities

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 simplicial complexes whose $f$-vectors are the $γ$-vectors in question. 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 inequalities. We conjecture that if $Δ$ is a flag homology sphere then $γ(Δ)$ satisfies the Kruskal-Katona inequalities. This conjecture is a significant refinement of Gal's conjecture, which asserts that such $γ$-vectors are nonnegative.

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