  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.
