Source authenticated

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.

Exact FrontierDelta

Prior state unknown→proved

Scope and record

Occurred: Sep 23, 2026

Delta type: SOURCE CLAIM

Assumptions: VibeMathed verification: lean-checked. Publication: announcement. AI contribution: ai-discovered. VibeMathed editorial classifications, scores, notes, relations, and dataset structure are CC BY 4.0. Source statements and linked content retain their own rights.

Canonical aliases: Seymour's second neighborhood conjecture

Confidence: Not scored

Registry verification: lean checked · announcement · candidate

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Attribution

VibeMathed
registry · event recorded by

Unreleased internal OpenAI model
model · ai model contributor · OpenAI

Evidence vector

SourceReplayReproducedFormal proofStatement auditExternal checkExpert reviewPeer review

Named evidence

VibeMathed record: Seymour's second neighborhood conjecture

specialist registry · passed

No independent mathematician has checked this yet. Checked here: abstract, introduction and Theorem 1.1 of the TeX source, read against the conjecture as recorded by Dean and Latka. Lean-checked on the release's Comparator challenge SeymourSecondNeighborhood (declaration OAI.SeymourSecondNeighborhood.exists_goodVertex, listed in lean/formalization.yaml). Its statement was read here: for every nonempty finite type with a loopless asymmetric relation, some vertex has at most as many first out-neighbours as second neighbours, where second neighbours exclude the vertex itself and its out-neighbours and are reached by a two-step path. That is exactly the headline claim, with no connectivity or degree hypothesis. Not rebuilt here. The minimal-counterexample and pruning argument was not refereed.

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Seymour's second neighborhood conjecture

witness · pending

Primary source named by the VibeMathed record. Presence in this import is not an independent mathematical verification.

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Lean Comparator challenge SeymourSecondNeighborhood (OpenAI math release)

lean artifact · pending

lean-proof link supplied by the VibeMathed registry.

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Artifacts and verifiers

Lean Comparator challenge SeymourSecondNeighborhood (OpenAI math release)

lean artifact · pending