Caccetta–Häggkvist conjecture

Let $r,g\in \mathbb{N}$. A digraph $D$ has girth at least $g$ if its shortest directed cycle has length at least $g$, and let $\delta^+(D)$ denote its minimum out-degree. Caccetta–Häggkvist conjecture. Every digraph $D$ with

$$\delta^+(D)\ge r$$

and girth at least $g$ contains at least

$$r(g-1)+1$$

vertices. This is a major open problem in extremal graph theory; it remains open even for $g=4$, and its $r$-regular special case is also unresolved.