Remarks on missing faces and generalized lower bounds on face numbers

TL;DR

This paper explores simplicial polytopes and complexes lacking missing faces above a certain dimension, proposing conjectures for lower bounds on face numbers and presenting partial results towards these conjectures.

Contribution

It introduces conjectures extending lower bounds on face numbers to complexes without high-dimensional missing faces and provides partial evidence for these conjectures.

Findings

Partial results supporting the conjectured bounds

Extension of classical lower bounds to new classes of complexes

Insights into the structure of complexes without missing faces

Abstract

We consider simplicial polytopes, and more general simplicial complexes, without missing faces above a fixed dimension. Sharp analogues of McMullen's generalized lower bounds, and of Barnette's lower bounds, are conjectured for these families of complexes. Some partial results on these conjectures are presented.