Dense-Case Progress on Seymour's Second Neighborhood Conjecture
Jake Brukhman proves Seymour's second neighborhood conjecture for oriented graphs with $n=2\delta+2$, raising the unconditional minimum order of a counterexample to 17; he reports directing and verifying work in which GPT-5-family models discovered the core counting argument and Claude models audited an intermediate draft.
In a manuscript dated August 9, 2026, Jake Brukhman proves Seymour's second neighborhood conjecture for every oriented graph on vertices, where is the minimum outdegree, without imposing a structure on the missing edges. Together with the known tournament case, this covers every oriented graph satisfying .
Seymour's conjecture asks for a vertex having at least as many vertices at exact outdistance two as immediate outneighbors. Brukhman's fixed-target counting argument implies that any counterexample has at least 17 vertices; conditional on a recent claimed result for minimum outdegree seven, this rises to 19. The manuscript explicitly notes that the method stops at , so the full conjecture remains open.
Brukhman states that he initiated and directed the investigation, curated intermediate results, selected the theorem, and edited the final presentation. OpenAI language models from the GPT-5 family reportedly performed detailed exploration, counterexample searches, verification tooling, discovery of the fixed-target argument, and manuscript drafting, while Anthropic Claude models adversarially audited an intermediate draft and assisted revisions. Brukhman says he verified the proofs and accepts responsibility for the final claims.