Intersecting faces of a simplicial complex via algebraic shifting

A family AA of sets is tt-intersecting   if the size of the intersection of every pair of sets in AA is at least tt, and it is an rr-family   if every set in AA has size rr. A well-known theorem of Erdős, Ko, and Rado bounds the size of a tt-intersecting rr-family of subsets of an nn-element set, or equivalently of (r−1)(r−1)-dimensional faces of a simplex with nn vertices. As a generalization of the Erdős–Ko–Rado theorem, Borg presented a conjecture concerning the size of a tt-intersecting rr-family of faces of an arbitrary simplicial complex. He proved his conjecture for shifted complexes. In this paper we give a new proof for this result based on work of Woodroofe. Using algebraic shifting we verify Borg’s conjecture in the case of sequentially Cohen–Macaulay ii-near-cones for t=it=i.