Title: The numbers of edges of 5-polytopes with a given number of vertices,
Abstract: 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 Grünbaum 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.