OpenAI · 170-02 · 2026-09-24 · Combinatorics
Sharp logarithmic exponents for fixed off-diagonal Ramsey numbers
preprints/Sharp-Logarithmic-Exponents-for-Fixed-Off-Diagonal-Ramsey-Numbers-September-24-2026
Family 170. Sharp logarithmic exponents for off-diagonal 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
For every fixed integer , proves as , determining the logarithmic exponent and matching the classical upper bound at that scale. Here is the least number of vertices forcing an -clique or a -vertex independent set.
For every fixed integer s ≥ 6, we determine the sharp logarithmic exponent of the off-diagonal Ramsey number:
Preprint source and Lean scope
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This preprint is named in the Lean scope note for 170: Sharp logarithmic exponents for off-diagonal 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.