Hirsch polytopes with exponentially long combinatorial segments

TL;DR

This paper introduces a local formulation of combinatorial segments in simplicial complexes, providing bounds on their lengths and demonstrating limitations through complex constructions, including Hirsch polytopes with exponentially long segments.

Contribution

It offers a new local definition of combinatorial segments, refines bounds for certain complexes, and constructs examples with exponential segment lengths, including Hirsch polytopes.

Findings

Bound of O(n2^d) for combinatorial segment length in normal complexes

Refined bound for banner complexes

Existence of complexes with segments of length Ω(n2^d)

Abstract

In their paper proving the Hirsch bound for flag normal simplicial complexes (Math. Oper.~Res.~2014) Adiprasito and Benedetti define the notion of~\emph{combinatorial segment}. The study of the maximal length of these objects provides the upper bound~$O(n2^d)$ for the diameter of any normal pure simplicial complex of dimension~$d$ with~$n$ vertices, and the Hirsch bound $n-d$ if the complexes are, moreover, flag. In the present article, we propose a formulation of combinatorial segments which is equivalent but more local, by introducing the notions of monotonicity and conservativeness of dual paths in pure simplicial complexes. We use this definition to investigate further properties of combinatorial segments. Besides recovering the two stated bounds, we show a refined bound for banner complexes, and study the behavior of the maximal length of combinatorial segments with respect to two usual operations, namely join and one-point suspension. Finally, we show the limitations of combinatorial segments by constructing pure normal simplicial complexes in which all combinatorial segments between two particular facets achieve the length $\Omega(n2^{d})$. This includes vertex-decomposable---therefore Hirsch---polytopes.