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

## TL;DR

This paper characterizes the possible pairs of the number of vertices and edges in 5-dimensional convex polytopes, extending known results from lower dimensions and confirming recent independent findings.

## Contribution

It provides a complete characterization of the first two entries of the $f$-vector for 5-polytopes, filling a gap in the combinatorial understanding of higher-dimensional polytopes.

## Key findings

- Characterization of vertex-edge pairs in 5-polytopes.
- Extension of Steinitz and Grünbaum's results to 5 dimensions.
- Independent confirmation of the characterization by other researchers.

## 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 Gr\"unbaum 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.