Erdős-Ko-Rado Type Theorems for Simplicial Complexes

Abstract

It is shown that every shifted simplicial complex $\Delta$ is EKR of type $(r,s)$, provided that the size of every facet of $\Delta$ is at least $(2s+1)r-s$. It is moreover proven that every $i$-near-cone simplicial complex is EKR of type $(r,i)$ if ${\rm depth}_{\mathbb{K}}\Delta\geq (2i+1)r-i-1$, for some field $\mathbb{K}$. Furthermore, we prove that if $G$ is a graph having at least $(2i+1)r-i$ connected components, including $i$ isolated vertices, then its independence simplicial complex $\Delta_G$ is EKR of type $(r,i)$. The results of this paper, generalize the main result of Frankl (2013).