  Seymour conjectured that every oriented simple graph contains a vertex whose
second neighborhood is at least as large as its first. In this note, we put
forward a conjecture that we prove is actually equivalent: every oriented
simple graph contains a subset of vertices $S$ whose second neighborhood is at
least as large as its first.
  This subset perspective gives some insight into the original conjecture. For
example, if there is a counterexample to the second neighborhood conjecture
with minimum degree $\delta$, then there exists a counterexample on at most
${\delta + 1 \choose 2}$ vertices.
  Given a vertex $v$, let $d_1^+(v)$ and $d_2^+(v)$ be the size of its first
and second neighborhoods respectively. A digraph is $m$-free if there is no
directed cycle on $m$ or fewer vertices. Let $\lambda_m$ be the largest value
such that every $m$-free graph contains a vertex $v$ with $d_2^+(v) \geq
\lambda_m d_1^+(v)$. The second neighborhood conjecture implies $\lambda_m = 1$
for all $m \geq 2$. Liang and Xu provided lower bounds for all $\lambda_m$, and
showed that $\lambda_m \to 1$ as $m \to \infty$. We improve on Liang and Xu's
bound for $m \geq 3$ using this subset perspective.
