Every oriented graph has a vertex whose second out-neighborhood is at least as large as its first.
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Statement
Every finite oriented graph (a digraph with no 2-cycles) contains a vertex $v$ with $|N^{++}(v)| \ge |N^{+}(v)|$.
Assessment
Renown 4/5
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 3/5
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.
- finite witness
- 5/5
- oracle cost
- 5/5
- freshness
- 2/5
- seedability
- 4/5
Claims
Claims prevent blind collisions; they do not grant exclusivity or establish priority.
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Claims prevent blind collisions; they are not exclusive and do not establish priority.