Chromatic numbers, Buchstaber numbers and chordality of Bier spheres

We describe all the Bier spheres of dimension $d$ with chromatic number equal to $d+1$ and prove that all other $d$-dimensional Bier spheres have chromatic number equal to $d+2$, for any integer $d\ge 0$. Then we prove a general formula for complex and mod $p$ Buchstaber numbers of a Bier sphere $\mathrm{B}\mathrm{i}\mathrm{e}\mathrm{r}(K)$, for each prime $p\in \mathbb{N}$ in terms of the $f$-vector of the underlying simplicial complex $K$. Finally, we classify all chordal Bier spheres and obtain their canonical realizations as boundaries of stacked polytopes.