Borg's EKR conjecture for simplicial complexes

Let $t\leq r$ be two natural numbers. A family of faces is $t$-intersecting if every pair of faces in it has intersection of cardinality at least $t$. Let $\Delta$ be a simplicial complex, let $\Delta_{(s)}$ denote its faces of cardinality $s$, and let $f_j(\Delta)$ be the number of faces of cardinality $j$. For a face $\sigma$ of $\Delta$, define its link by

$$\operatorname{lk}_{\Delta}\sigma=\{\tau:\tau\cup\sigma\in\Delta,\ \tau\cap\sigma=\emptyset\}.$$

Assume that the minimal facet cardinality of $\Delta$ is $k\geq(t+1)(r-t+1)$, and let $S\neq\emptyset$ be a subset of $[t,r]$. Borg's EKR conjecture. Every $t$-intersecting family $\mathcal{A}$ of faces of $\Delta$ satisfying

$$\mathcal{A}\subseteq\bigcup_{s\in S}\Delta_{(s)}$$

obeys

$$|\mathcal{A}|\leq\max_{\sigma}\sum_{s\in S}f_{s-t}(\operatorname{lk}_{\Delta}\sigma),$$

where the maximum is taken over all $t$-faces $\sigma$ of $\Delta$. This asserts that a family consisting of all faces in the specified cardinalities containing a suitable fixed $t$-face has maximal size among such $t$-intersecting families. The conjecture generalizes the Erdős–Ko–Rado theorem from uniform set families to faces of simplicial complexes and allows several face cardinalities simultaneously. Its resolution status is not supplied in the source.