Erdo{double acute}s-Ko-Rado theorems for simplicial complexes

摘要

A recent framework for generalizing the Erdo{double acute}s-Ko-Rado theorem, due to Holroyd, Spencer, and Talbot, defines the Erdo{double acute}s-Ko-Rado property for a graph in terms of the graph's independent sets. Since the family of all independent sets of a graph forms a simplicial complex, it is natural to further generalize the Erdo{double acute}s-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.