Holroyd–Talbot conjecture on the EKR property of graphs

Let $G$ be a graph, let $calI^{(r)}(G)$ denote the family of independent $r$-subsets of $V(G)$, and let $calI^{(r)}_v(G)$ denote the star centered at $v\in V(G)$. Write $cmu(G)$ for the size of a smallest maximal independent set of $G$. A graph is $r$-EKR if there is a vertex $v\in V(G)$ such that every intersecting family $calF\subseteq calI^{(r)}(G)$ satisfies $|calF|\leq |calI^{(r)}_v(G)|$. Holroyd–Talbot conjecture. Every graph $G$ is $r$-EKR whenever

$$1\leq r\leq \frac{\mu(G)}{2}.$$

This conjecture extends the Erdős–Ko–Rado theorem from uniform set systems to independent sets in graphs. It asserts that the threshold for the EKR property is governed by the minimum size of a maximal independent set; its general status is not specified in the source.