On vertex-minimal simplicial maps to the sphere

For positive integers $n,d$, let $\lambda (n,d)$ be the minimal number of vertices of a triangulation of $n$-sphere which admits a degree $d$ simplicial map to the boundary of $(n+1)$-simplex. We show that ${\lim }_{d\to \mathrm{\infty }}\frac{\lambda (n,d)}{d}=0$ for any $n\ge 3$, disproving O. Musin’s conjecture. Using similar idea, for any $C$ we construct a triangulation of ${\mathbb{S}}^n$, $n\ge 3$, for which $\frac{f_j}{f_i}>C$, for any $0\le i<j\le n$ such that $i<\lfloor \frac{n-1}{2}\rfloor$. All triangulations we obtain are isomorphic to boundaries of convex polytopes in ${\mathbb{R}}^{n+1}$.