Characterizing face and flag vector pairs for polytopes

TL;DR

This paper provides a complete characterization of certain face and flag number pairs of 4-dimensional polytopes, extending known results and analyzing exceptional cases for higher dimensions.

Contribution

It introduces a full characterization of flag number pairs for 4-polytopes and describes face number pairs for higher dimensions, including exceptional cases.

Findings

Complete characterization of $(f_0,f_{03})$ for 4-polytopes.

Description of $(f_0,f_{d-1})$ pairs for even and odd dimensions.

Identification of finitely many small exceptional pairs in higher dimensions.

Abstract

Gr\"unbaum, Barnette, and Reay in 1974 completed the characterization of the pairs $(f_i,f_j)$ of face numbers of $4$-dimensional polytopes. Here we obtain a complete characterization of the pairs of flag numbers $(f_0,f_{03})$ for $4$-polytopes. Furthermore, we describe the pairs of face numbers $(f_0,f_{d-1})$ for $d$-polytopes; this description is complete for even $d\ge6$ except for finitely many exceptional pairs that are "small" in a well-defined sense, while for odd $d$ we show that there are also "large" exceptional pairs. Our proofs rely on the insight that "small" pairs need to be defined and to be treated separately; in the $4$-dimensional case, these may be characterized with the help of the characterizations of the $4$-polytopes with at most $8$ vertices by Altshuler and Steinberg (1984).