Erdős–Ko–Rado theorems for simplicial complexes

A recent framework for generalizing the Erdős–Ko–Rado theorem, due to Holroyd, Spencer, and Talbot, defines the Erdős–Ko–Rado property for a graph in terms of the graphʹs independent sets. Since the family of all independent sets of a graph forms a simplicial complex, it is natural to further generalize the Erdős–Ko–Rado property to an arbitrary simplicial complex. An advantage of working in simplicial complexes is the availability of algebraic shifting, a powerful shifting (compression) technique, which we use to verify a conjecture of Holroyd and Talbot in the case of sequentially Cohen–Macaulay near-cones.