The EKR property for flag pure simplicial complexes without boundary

TL;DR

This paper proves that flag pure simplicial complexes without boundary ridges satisfy the Erdős-Ko-Rado property up to dimension three, and conjectures this extends to higher dimensions, with implications for flag pseudo-manifolds and cluster complexes.

Contribution

It establishes the EKR property for flag pure simplicial complexes without boundary ridges up to dimension three and proposes a conjecture for all dimensions.

Findings

EKR property holds for dimensions up to three

Conjecture extends the result to arbitrary dimensions

Includes flag pseudo-manifolds and cluster complexes

Abstract

We prove that the family of facets of a pure simplicial complex of dimension up to three satisfies the Erd\H{o}s-Ko-Rado property whenever it is flag and has no boundary ridges. We conjecture the same to be true in arbitrary dimension and give evidence for this conjecture. Our motivation is that complexes with these two properties include flag pseudo-manifolds and cluster complexes.