The numbers of edges of 5-polytopes with a given number of vertices

A basic combinatorial invariant of a convex polytope $P$ is its $f$-vector $f(P)=(f_0,f_1,dots,f_{dim P-1})$, where $f_i$ is the number of $i$-dimensional faces of $P$. Steinitz characterized all possible $f$-vectors of $3$-polytopes and Grunbaum characterized the pairs given by the first two entries of the $f$-vectors of $4$-polytopes. In this paper, we characterize the pairs given by the first two entries of the $f$-vectors of $5$-polytopes. The same result was also proved by Pineda-Villavicencio, Ugon and Yost independently.