Seymour's Second Neighborhood Conjecture
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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 .
Context
A dense case, not the conjecture: it remains open in general. The concrete gain is on the size of any counterexample - combined with the known minimum-outdegree results, this raises the best known lower bound on the order of a counterexample from 16 to 17, and to 19 conditional on the 2026 preprint of Sadhukhan, Sandeep and Sen. The novelty against the earlier dense-case work is that no structure is prescribed on the missing edges.
Seymour's second neighborhood conjecture is a well-known problem in digraph theory, open since around 1990 and the subject of a continuing literature - Fisher's tournament theorem, the minimum-outdegree results of Kaneko and Locke, and a decade of dense-case work by Fidler-Yuster, Ghazal and Dara-Francis-Jacob-Narayanan. Scored for the problem rather than this increment, level with the other well-tracked named conjectures and below the household ones.
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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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