Intersecting faces of a simplicial complex via algebraic shifting

A family A of sets is t-intersecting if the size of the intersection of every pair of sets in A is at least t, and it is an r-family if every set in A has size r. A wellknown theorem of Erdős, Ko, and Rado bounds the size of a t-intersecting r-family of subsets of an n-element set, or equivalently of (r−1)-dimensional faces of a simplex with n vertices. As a generalization of the Erdős-Ko-Rado theorem, Borg presented a conjecture concerning the size of a t-intersecting r-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 i-near-cones for t = i.