OpenAI · 189-01 · 2026-09-25 · Combinatorics
Cycle--clique Ramsey numbers
preprints/Cycle-clique-Ramsey-numbers-September-25-2026
Family 189. Exact cycle–clique Ramsey numbers.
The authors are OpenAI. This companion does not verify the proof and does not adopt it as a theorem of Ngo Cheung’s series. It is not a premise. O.1 stays NOT_CLOSED. Anti-de Sitter results are not a de Sitter state, not Em02-E’s relational lapse, not XVI-P’s QdS scheme, and not XVI-Q’s sigma-image record. No quantum coefficient is admitted. No compilation is asserted.
Read against the standing problems
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
Proves the Erdős–Faudree–Rousseau–Schelp conjecture: for every , except . This is the exact threshold forcing a red -cycle or a blue -clique in every red–blue coloring of a complete graph.
We prove that for every pair of integers other than , for which . This establishes the cycle–clique conjecture of Erdős, Faudree, Rousseau and Schelp. The proof combines expansion in a minimal counterexample with a large-clique lemma and an optimization of paths joining clique vertices. These arguments reduce the remaining cases to finite parameter-pattern instances, which are excluded by two exact implementations of proved inference rules. Complete programs and deduction traces accompany the paper.
Preprint source and Lean scope
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This preprint is named in the Lean scope note for 189: Cycle–clique Ramsey numbers. The authors are OpenAI. This companion does not verify the formalization. It is not a premise. The OAI library is not imported. No compilation is asserted. O.1 stays NOT_CLOSED.