Abstract
Eberhard proved that for every sequence $(p_k), 3\le k\le r, k\ne 6$ of non-negative integers satisfying Euler's formula $\sum_{k\ge3} (6-k) p_k = 12$, there are infinitely many values $p_6$ such that there exists a simple convex polyhedron having precisely $p_k$ faces of size $k$ for every $k\ge3$, where $p_k=0$ if $k>r$. In this paper we prove a similar statement when non-negative integers $p_k$ are given for $3\le k\le r$, except for $k=5$ and $k=7$ (but including $p_6$).
We prove that there are infinitely many values $p_5,p_7$ such that there exists a simple convex polyhedron having precisely $p_k$ faces of size $k$ for every $k\ge3$. We derive an extension to arbitrary closed surfaces, yielding maps of arbitrarily high face-width.
Our proof suggests a general method for obtaining results of this kind.