Seymour's second neighborhood conjecture

Theorem 1.1: every nonempty finite oriented graph has a vertex $v$ with $|N_1^+(v)|\le|N_2^+(v)|$, with no connectivity, density or degree assumption. Via Seacrest's equivalences it also gives the vertex- and arc-weighted forms, and it yields short-cycle consequences, for example a directed triangle whenever minimum in- and outdegree are at least one third of the order. It does not address infinite digraphs or digraphs with 2-cycles, which the conjecture excludes.

Theorem 1.1: every nonempty finite oriented graph has a vertex $v$ with $|N_1^+(v)|\le|N_2^+(v)|$, with no connectivity, density or degree assumption. Via Seacrest's equivalences it also gives the vertex- and arc-weighted forms, and it yields short-cycle consequences, for example a directed triangle whenever minimum in- and outdegree are at least one third of the order. It does not address infinite digraphs or digraphs with 2-cycles, which the conjecture excludes.