  We present examples of flag homology spheres whose $\gamma$-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 $\gamma$-vectors in
question. In another direction, we show that if a flag $(d-1)$-sphere has at
most $2d+2$ vertices its $\gamma$-vector satisfies the Kruskal-Katona
inequalities. We conjecture that if $\Delta$ is a flag homology sphere then
$\gamma(\Delta)$ satisfies the Kruskal-Katona inequalities. This conjecture is
a significant refinement of Gal's conjecture, which asserts that such
$\gamma$-vectors are nonnegative.
