  Consider the question: Given integers $k<d<n$, does there exist a simple
$d$-polytope with $n$ faces of dimension $k$? We show that there exist numbers
$G(d,k)$ and $N(d,k)$ such that for $n> N(d,k)$ the answer is yes if and only
if $n\equiv 0\quad \pmod {G(d,k)}$. Furthermore, a formula for $G(d,k)$ is
given, showing that e.g. $G(d,k)=1$ if $k\ge
\left\lfloor\frac{d+1}{2}\right\rfloor$ or if both $d$ and $k$ are even, and
also in some other cases (meaning that all numbers beyond $N(d,k)$ occur as the
number of $k$-faces of some simple $d$-polytope).
  This question has previously been studied only for the case of vertices
($k=0$), where Lee \cite{Le} proved the existence of $N(d,0)$ (with $G(d,0)=1$
or $2$ depending on whether $d$ is even or odd), and Prabhu \cite{P2} showed
that $N(d,0) \le cd\sqrt {d}$. We show here that asymptotically the true value
of Prabhu's constant is $c=\sqrt2$ if $d$ is even, and $c=1$ if $d$ is odd.
