Converse of Euler's formula for polyhedra

Question

Is it true that for any positive integers $V, E, F$ with $V - E + F = 2$ there exists a polyhedron with $V$ vertices, $E$ edges and $F$ faces?

In case there is a silly counterexample (say, with $F=1$), then what about large $V,E,F$ - say, all greater than or equal to $10$?

Any help appreciated!

Answer 1

For any $F$ there exist $V$ and $E$ with $V-E+F=2$ but no polyhedron with $V$ vertices, $E$ edges, and $F$ faces. Indeed, the number of edges which can fit on the $F$ polygons is finite and hence we can take $E$ much larger and define $V:=2-F+E$.

Answer 2

There is no polyhedron $P\subset{\mathbb R}^3$ with $7$ edges. This is well known, and is proven, e.g., here as an exercise: https://people.math.ethz.ch/~blatter/Mathesis.pdf