##Eberhard Theorem

Consider a simple 3-polytope P, (so every vertex has 3 neighbors). If $p_k$ is the number of faces of P which are k-gonal, Euler's theorem implies that $\sum_{k \ge 3} (6-p_k)=12$. Note that 6-gonal faces do not contribute to the LHS. (One way to think about it is that polygonal faces with 7 and more sides contributes "negative curvature", small faces contributes positive "curvature" and hexagons are "flat".)

Eberhard's theorem asserts that if you have a sequence of numbers $p_k, k \ne 6$ such that $\sum_{k \ge 3} (6-p_k)=12$ then you can find a simple 3-polytope with $p_k$ k-gonal faces. (But you have no control on $p_6$). 

##Extensions of Eberhard theorem

There are various results extending Eberhard's theorem in various directions. Chapter 13 in Grunbaum's book "[Convex Polytopes][1]" and especially the supplemantary material at the end of the chapter in the new 2nd edition is a good source. Another general source is [my chapter][2] from the "Handbook of Discrete and Computational geometry" on garphs and skeleta of polytops.  


A relatively recent paper on the subject is by Stanislav Jendrol "[On the face vectors of trivalent convex polyhedra][3]". Another paper by Jendrol which deals with general 3-polytopes from the same year is "On face vectors and vertex vectors of convex polyhedra" Discrete Math 118 (1993)119-144. There are analogs of Eberhard theorem for 4-regular planar graphs, for toroidal graphs and in other directions.

A far as I know, there is no good answer known for the question posed by Shephard of characterizing all sequences $(p_3,p_4,\dots)$, and no such characterization is known even for the simple case. 

##High dimensions

In higher dimensions and even in four dimensions these problems become very difficult and very little is known? You can ask again about the numbers of k-gonal 2-dimensional faces. While the formula above implies that in dimension 3 and more $p_3+p_4+p_5>0$ it is known that in dimensions 5 and more $p_3+p_4>0$. 

Perhaps an even more natural extension is to consider the type of facets a given d-simensional polytope have. You can ask for a simple 4 polytope (a 4-polytope whose graph is 4-regular) what are the number of facets $p_Q$ isomorphic to a given 3-polytope Q. This gives you a vector indexed by combinatorial types of simple 3-polytopes, but I am not aware of any Eberhard type theorem and I do not know even which 3-polytopes should be considered as the analogs of the hexagons in the above formula. Dually stated and extented to triangulations of 3-spheres the question is to associate to a triangulated 3-simensional sphere (or just simplicial 3-polytope) the list of links of vertices (with multiplicities) it has. A related MO question is [this one][4].

There are some reasons to jump from dimension 3 directly to dimension 5. The nature of the problem is somewhat different in even and odd dimensions. I will try to elaborate on that in the next edit.


  [1]: http://books.google.co.il/books?id=ISHO86XJ1CsC&printsec=frontcover&dq=grunbaum's+convex+polytopes&source=bl&ots=S7T41O6hvF&sig=wRkbvA0TtPVeRuiRFHo6FjVQv1I&hl=iw&ei=QkrqS7LzCaOiOLaTsPMK&sa=X&oi=book_result&ct=result&resnum=1&ved=0CAgQ6AEwAA#v=onepage&q=grunbaum's%2520convex%2520polytopes&f=false
  [2]: http://www.ma.huji.ac.il/~kalai/ch19.pdf
  [3]: http://pdf.dml.cz/bitstream/handle/10338.dmlcz/136327/MathSlov_33-1983-2_7.pdf
  [4]: http://mathoverflow.net/questions/10503/simplicial-and-cubical-decompositions-of-low-valence