The Erd\H{o}s-Ko-Rado property of trees of depth two

Abstract

A family of sets is intersecting if any two sets in the family intersect. Given a graph $G$ and an integer $r\geq 1$, let $\mathcal{I}^{(r)}(G)$ denote the family of independent sets of size $r$ of $G$. For a vertex $v$ of $G$, let $\mathcal{I}^{(r)}_v(G)$ denote the family of independent sets of size $r$ that contain $v$. This family is called an $r$-star. Then $G$ is said to be $r$-EKR if no intersecting subfamily of $ \mathcal{I}^{(r)}(G)$ is bigger than the largest $r$-star. Let $k, n, r \geq 1$, and let $T(n, k)$ be the tree of depth two in which the root has degree $n$ and every neighbour of the root has the same number $k + 1$ of neighbours. For each $k \geq 2$, we show that $T(n, k)$ is $r$-EKR if $2r \leq n$, extending results of Borg and of Feghali, Johnson and Thomas who considered the case $k = 1$.