Erdos-Ko-Rado theorems for simplicial complexes

TL;DR

This paper extends the Erdos-Ko-Rado theorem to simplicial complexes, utilizing algebraic shifting to verify a conjecture for specific classes of complexes, thereby broadening the theorem's applicability.

Contribution

It generalizes the Erdos-Ko-Rado property from graphs to arbitrary simplicial complexes and applies algebraic shifting to verify a key conjecture in this context.

Findings

Verified a conjecture of Holroyd and Talbot for sequentially Cohen-Macaulay near-cones.

Demonstrated the usefulness of algebraic shifting in studying combinatorial properties of simplicial complexes.

Extended the Erdos-Ko-Rado theorem to a broader mathematical setting.

Abstract

A recent framework for generalizing the Erdos-Ko-Rado Theorem, due to Holroyd, Spencer, and Talbot, defines the Erdos-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 Erdos-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.