Missing faces of neighborly and nearly neighborly polytopes and spheres

Abstract: For a $(d-1)$-dimensional simplicial complex $\Delta$ and $1\leq i\leq d$, let $f_{i-1}$ be the number of $(i-1)$-faces of $\Delta$ and $m_i$ be the number of missing $i$-faces of $\Delta$. In the nineties, Kalai asked for a characterization of the $m$-numbers of simplicial polytopes and spheres -- a problem that remains wide open to this day. Here, we study the $m$-numbers of nearly neighborly and neighborly polytopes and spheres. Specifically, for $d\geq 4$, we obtain a lower bound on $m_{\lfloor d/2\rfloor}$ in terms of $f_0$ and $f_{\lfloor d/2\rfloor-1}$ in the class of all $(\lfloor d/2\rfloor-1)$-neighborly $(d-1)$-spheres. For neighborly spheres, we (almost) characterize the $m$-numbers of $2$-neighborly $4$-spheres, and we show that, for all odd values of $k$, there exists an infinite family of neighborly simplicial $2k$-spheres with $m_{k+1}=0$. Along the way, we provide a simple numerical condition based on the $m$-numbers that allows to establish non-polytopality of some neighborly odd-dimensional spheres.