Seymour's second neighbourhood conjecture

Let $G$ be a directed graph. For a vertex $v$, write $N^+_1(G,v)$ for its out-neighbourhood and $N^+_2(G,v)$ for the vertices reachable from $v$ in two steps but not in one step. An orientation is a directed graph with no directed cycles of length two.

Seymour's second neighbourhood conjecture. Every orientation $G$ contains at least one vertex $v$ such that

$$|N^+_2(G,v)| \geq |N^+_1(G,v)|.$$

This is a central open problem about second neighbourhoods in oriented graphs; the paper studies the equality case through Seymour-tight orientations.