I think there is a theorem with this statement, I'd like to know its name if it has a common one.
Also I was reading a proof for this theorem, it stated that:
Suppose that $c<\alpha (D)$ and let $P_1, \cdots , P_c$ pairwisely disjoint directed paths covering $D$. Let $S$ be a maximum stable, then there exist $i$ such that $|P_i \cap S| \ge 2 \cdots$
But I didn't understand this step, why should there exist an $i$ with $|P_i \cap S| \ge 2$ ??
(Note that $c$ is considered as the minimum number of pairwisely disjoint paths covering $D$)
Any explanation (and completion) of this proof is appreciated, or if there is another proof its okay.