The birational geometry of the moduli space of stable, n-pointed, rational curves is still largely unknown. For example, it is not known if the cone of effective divisors is polyhedral, if the cone of effective curves is generated by 1-strata (Fultons conjecture), or if it has only finitely many small modifications. In a joint work with Jenia Tevelev, we find that some of this complexity can be explained by the fact that this moduli space can be interpreted as the Brill-Noether locus of some very special reducible curves related to hypergraphs. Using this we construct many new extremal divisors and birational morphisms with unexpected exceptional loci.