An Erd\H{o}s-Ko-Rado Theorem for unions of length 2 paths

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 $G$, the family of independent sets of size $r$ that contain $v$ 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 $n$ be a positive integer, and let $G$ consist of the disjoint union of $n$ paths each of length 2. We prove that if $1 \leq r \leq n/2$, then $G$ is $r$-EKR. This affirms a longstanding conjecture of Holroyd and Talbot for this class of graphs and can be seen as an analogue of a well-known theorem on signed sets, proved using different methods, by Deza and Frankl and by Bollob\'as and Leader. Our main approach is a novel probabilistic extension of Katona's elegant cycle method, which might be of independent interest.