Seymour's Second Neighborhood Conjecture
Statement
Seymour conjectured that every finite oriented graph has a vertex with at least as many exact second outneighbors as outneighbors. Known cases include tournaments (Fisher 1996) and minimum outdegree at most six (Kaneko-Locke 2001), and for dense incomplete graphs a series of results restricting the structure of the missing edges. This work proves the conjecture for every oriented graph of order , where is the minimum outdegree, with no prescribed structure on the missing edges; with Fisher's tournament theorem this gives every oriented graph satisfying .
Record
Comments
No person has examined this. Nothing here has been checked at all. say whether it holds →
proof attempt · #1
GPT-5 family, Claude, with Jake BrukhmanThe record says a model found this and names the people who worked on it. No ProbXiv account is credited for it, and nobody has answered for it here.
The acknowledgements are unusually specific about the division of labour: "The author initiated and directed the investigation, curated intermediate results, selected the theorem for publication, and edited the final statement and exposition. OpenAI language models (GPT-5 family) carried out the detailed mathematical exploration, implemented counterexample searches and verification tools, discovered the fixed-target capacity argument and its double-counting proof, and drafted the manuscript; Anthropic Claude models performed an adversarial audit of an intermediate draft and assisted with revisions. The author verified the proofs and accepts sole responsibility for the final manuscript and its claims." The model is credited with discovering the central argument by name, which is the discovered tier rather than the co-developed one.
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