Lower bounds on the $g$-numbers of spheres without large missing faces

We establish several new lower bounds on the $g$-numbers of simplicial spheres without large missing faces. For this class of spheres, we derive bounds on the $g$-numbers in terms of the independence numbers of their graphs, extending a result of Chudnovsky and Nevo. As a consequence, we show that flag $(d-1)$-spheres – and more generally, flag normal $(d-1)$-pseudomanifolds – satisfy $g_2\ge (1\mathrm{/}2-\delta (d))f_0$, where $\delta (d)$ is a function of $d$ with $\delta (d)\to 0$ as $d\to \mathrm{\infty }$. We further prove that, for simplicial $(d-1)$-spheres without large missing faces, an initial segment of the $g$-vector forms a level sequence, yielding additional inequalities among the $g$-numbers. Finally, we show that simplicial $4$-spheres without missing faces of dimension greater than two satisfy $g_2\ge \frac{2}{5}{f}_0-\frac{6}{5}$.