Title: Strict Erdős-Ko-Rado Theorems for Simplicial Complexes
Abstract: The now classical theorem of Erdős, Ko and Rado establishes
the size of a maximal uniform family of pairwise-intersecting sets as well
as a characterization of the families attaining such upper bound.
One natural extension of this theorem is that of restricting the possible
choices for the sets. That is, given a simplicial complex,
what is the size of a maximal uniform family of pairwise-intersecting faces.
Holroyd and Talbot, and Borg conjectured that the same phenomena as
in the classical case (i.e., the simplex) occurs: the maximal
such family is given by all the faces having some common vertex.
Moreover, they also conjectured that if a family attains such bound then its
members must have a common vertex. In this talk I will present some
progress towards the characterization of such maximal families. Concretely
I will show that the conjecture is true for near-cones of sufficiently
high depth. In particular, this implies that the characterization
of maximal families holds for, for example, the independence complex
of a chordal graph with an isolated vertex as well as the independence
complex of a (large enough) disjoint union of graphs with at least one
isolated vertex. Under stronger hypothesis, i.e., more isolated vertices,
we also recover a stability theorem.
This talk is based on a joint work with Russ Woodroofe.
Preprint: arXiv:2503.15608