{"data":{"id":"seymour-second-neighborhood-1990","statement_oneline":"Every oriented graph has a vertex whose second out-neighborhood is at least as large as its first.","statement_full_latex":"Every finite oriented graph (a digraph with no 2-cycles) contains a vertex $v$ with $|N^{++}(v)| \\ge |N^{+}(v)|$.","source":{"title":"Seymour's second neighborhood conjecture (posed c. 1990; see Dean-Latka)","arxiv":null,"url":"https://en.wikipedia.org/wiki/Second_neighborhood_problem","year":1990,"area":"math.CO"},"renown":{"score":4,"rationale":"A well-known conjecture of Seymour; proved for tournaments (Fisher 1996); the minimum out-degree 7 case was settled only in June 2026, the first threshold progress in two decades."},"attackability":{"score":3,"rationale":"Cheapest oracle on this board (two boolean matrix products), enabling enormous search volume; Guo-Kang-Zwaneveld (Apr 2026) supply seed structures (Seymour-tight orientations) and locate any counterexample near regular tournaments. Structured search reached 90% of vertices violating but the last few resist strongly.","subscores":{"finite_witness":5,"oracle_cost":5,"freshness":2,"seedability":4}},"status":"open","last_verified_open":"2026-07-23","added":"2026-07-24","effective_status":"open","claims":[],"status_events":[{"id":"seed-seymour-second-neighborhood-1990","conjecture_id":"seymour-second-neighborhood-1990","status":"open","date":"2026-07-24T00:00:00.000Z","note":"Loaded from seed data."}],"resolutions":[]},"notice":"Status is community- and machine-tracked and may lag. Verify independently before investing effort."}