The pure-EKR conjecture for flag simplicial complexes without boundary

A simplicial complex is pure if all its facets have the same cardinality, flag if every set of pairwise adjacent vertices is a face, and without boundary if every non-maximal simplex is contained in at least two facets. A pure simplicial complex is pure-EKR if the family of its facets has the Erdős–Ko–Rado property, meaning that every intersecting family of facets has size at most the size of the star of some vertex.

Pure-EKR conjecture. Every pure, flag simplicial complex without boundary is pure-EKR.

This extends the polygon-triangulation conjecture and includes flag pseudo-manifolds and cluster complexes. The paper proves the assertion for complexes of dimension at most three, while the arbitrary-dimensional case remains open.